FRANK'I WARNER
and Comp
Glmie illimis Lomlm

This textbook fills a long-standing gap. The beginning graduate student
finds it hard to learn the basic material on differentiable manifolds. All the
books he is referred to give a cursory treatment and quickly move on to more
specialized topics. For this reason, Professor Warner's book is especially
welcome. Here is a clear, detailed, and careful development of the funda-
mental facts on manifold theory and Lie groups. Numerous problems extend
the theory and help the student master the subject. An added bonus is the
sheaf-theoretic proof of the de Rham theorem and an elementary proof of
the Hodge theorem. As far as I know, the latter proof is the only one in the
literature easily accessible to the novice in analysis.
I. NI. inger

This book provides the neemsary foundation for students interested in any
of the diverse areas of mathematics which require the notion of a differentiable
manifold. It is designed as a beginning graduate-level textbook and presumes
a good undergraduate training in algebra and analysis plus som knowledge
of point set topology, covering spaces, and the fundamental group. It is
also intended for use as a reference book since it includes a number of items
which are difficult to ferret out of the literature, in particular, thecompleteand
self-contained proofs of the fundamental theorems of Hodge and de Rham.
The core material is contained in Chapters 1, 2, and 4. This includes
difl'erentiable manifolds, tangent vectors, submanifolds, implicit function
theorems, vector fields, distributions and the Frobenius theorem, differential
forms, integration, Stokes' theorem, and de Rham eohomology.
Chapter 3 treats the foundations of Lie group theory, including the
relationship between Lie groups and their Lie algebras, the exponential
map, the adjoint representation, and the closed subgroup theorem. Many
examples are given, and many properties of the classical groups are derived.
The chapter concludes with a discussion of homogeneous manifolds. The
standard reference for Lie group theory for over two decades has been
Chevalley's Theory of Lie Groups, to which I am greatly indebted.
For the de Rham theorem, which is the main goal of Chapter 5, axiomatic
sheaf cohomology theory is developed. In addition to a proof of the strong
form of the de Rham theorem--the de Rham homomorphism given by
integration is a ring isomorphism from the de Rham cohomology ring to the
differentiable singular cohomology ring--it is proved that there are canonical
isomorphisms of all the classical cohomology theories on manifolds. The
pertinent pans of all these theories are developed in the text. The approach
which I have followed for axiomatic sheaf cohomology is due to H. Cartan,
who gave an exposition in his Sdminaire 1950/1951.
For the Hodge theorem, a complete treatment of the local theory of
elliptic operators is presented in Chapter 6, using Fourier series as the basic
tool. Only a slight acquaintance with Hilbert spaces is presumed.
I wish to thank Jerry Kazdan, who spent a large portion of the
summer of 1969 educating me to the whys and wherefores of inequalities
and who provided considerable assistance with the preparation of this chapter.
I also benefited from notes on lectures by J. J. Kohn and Stephen Andrea,
from several papers of Louis Nirenberg, and from Partial Differential
-\

F_uations by Bets, John, and Scbechter, which the reader might wish to
consult for further references to the literature.
At the end of each chapter is a set of exercises. These are an integral
part of the text. Often where a claim in a chapter has been left to the reader,
there is a reminder in the Exercises that the reader should provide a proof
of the edaim. Some exercises are routine and test general understanding
of the chapter. Many present significant extensions of the text. In some
cases the exercises contain major theorems. Two notable examples are
properties of the eigenfunctions of the Laplacian and the Peter-Weyl theorem,
which are developed in the Exercises for Chapter 6. Hints are provided for
many of the difcult exercises.
There are a few notable omissions in the text. I have not treated complex
manifolds, although the sheaf theory developed in Chapter 5 will provide
the reader with one of the basic tools for the study of complex manifolds.
Neither have I treated infinite dimensional manifolds, for which I refer the
reader to Lang's Introduction to Differentiable Manifold, nor Sard's theorem
and imbedding theorems, which the reader can find in Sternberg's Lectures
on Differential Geometr)'.
Several possible courses can be based on this text. Typical one-semester
courses would cover the core material of Chapters 1, 2, and 4, and then
either Chapter 3 or 5 or 6, depending on the interests of the class. The
entire text can be covered in a one-year course.
Students who wish to continue with further study in differential geometry
should consult such advanced texts as Differential Geometry and Symmetric
Spaces by Helgason, Geometry of Manifolds by Bishop and Crittenden, and
Foundations of Differential Geometry (2 vols.) by Kobayashi and Nomizu.
I am happy to express my gratitude to Professor I. M. Singer, from whom
I learned much of the material in this book and whose courses have always
generated a great excitement and enthusiasm for the subject.
Many people generously devoted considerable time and effort to reading
early versions of the manuscript and making many corrections and helpful
suggestions. I particularly wish to thank Manfredo do Carmo, Jerry Kazdan,
Stuart Newberger, Marc Rieffel, John Thorpe, NolanWailach, Hung-Hsi Wu,
and the students in my classes at the University of California at Berkeley and
at the University of Pennsylvania. My special thanks to Jeanne Robinson..
Marian Griflths, and Mary Ann Hippie for their excellent job of typing, and
to Nat Weintraub of Scott, Foresman and Company for his cooperation and
 excellent guidance and assistance in the final preparation of the manuscript.
Frank Warner

2 Preliminarim
$ Diffrenfiabl Manifolds
$ The Second Axiom of Countabil/ty
• II Tangent Vctors and Differentials
22 Submanifolds, Diffeomorphisms, ud the Inverse Function Theorem
0 Impl/cit Function Theorems
4 Vector Fields
4| Distributions and the Frobcnius Theorem
0 Exerrses
 TENSORS AND DIFIRENTIAL FORMS
4 Tensor ud Exterior Algebras
62 Tensor Fields and Differential Forms
69 The Lie Derivative
75 Differential Ideals
77 Exerc/ses
GROUPS
82 Li Groups ud Their Lie Algebras
89 Homomorphisms
92 Lie Subgroups
98 Coveringe
|01 Simply Connoctod Lie Groups
102 Exponential Map
109 Continuous Homomorphisms
110 Closed Subgroups
112 The Adjoint Representation
117 Automorphisms and Derivations of Bilinear Operations and Forms
120 Homogeneous Manifolds
152 Exercises

INTGRATIOI ON MANIFOLDS
1 $8 Orientation
140 Integration on Manifolds
155 de Rham Cohomology
157 Exercises
SHEAVES, COHOMOLOGY, AND THE DE R.HAM
THEOREM
165 Sheaves and Presheaves -
175 Cozhain Complexes
176 Axiomatic Sheaf Cohomology
The Classical Cohomology Theories
186 Alexander-Spanier Cohomolog),
189 de Rham Cohomology
191 Singular Cohomology
200 ech Cohomology
205 The de Rham Theorem
207 Multiplicative Structure
214 Supports
216 Exercises
220
222
227
240
243
250
251
THE HODGE THEOREM
The Laplace-Beltrami Operator
The Hodge Theorem
Some Calculus
Elliptic Operators
Reduction to the Periodic Case
Ellipticity of the Laplace-Beltrami Operator
Exercises
26O BIBLIOGRAPHY
262 INDEX OF NOTATION
265 INDEX

After establishing some notational conventions which will be used throughout
the book, we will begin with the notion of a differentiable manifold. These
are spaces which are locally like Euclidean space ud which have enough
structure so that the basic concepts of calculus can be curied over. In this
first chapter we shall primarily be concerned with the analogs ud implications
for manifolds of the fundamental theorems of differential calculus. Later,
in Chapter 4, we shall consider the theory of integration on manifolds.
From the notion of directional derivative in Euclidean space we will
obtain the notion of a tangent vector to a differentiable manifold. We will
study mappings between manifolds ud the effect that toppings have on
tangent vectors. We will investigate the impfications for mappings of
manifolds of the classical inverse ud implicit function theorems. We will
see that the fundamental existence and uniqueness theorems for ordinary
differential equations translate into existence ud uniqueness statements for
integral curves of vector fields. The chapter doses with the Frobenius
theorem, which pertains to the existence and uniqueness of integral manifolds
of involutive distributions on manifolds.
PRELIMINARIES
1.1 Some Basic Notation and Term/nolosy Throughout this text we
will describe sets either by listings of their elements, for example
{a,..., a.},
or by expressions of the form
which denote the set of all x satisfying property P. The expression a  A
meam that a is an element of the set A. If a set A is a subset of a set B (that
is, a  B whenever a  A), we write A  B. If A = B and B = A, then A
equal B, denoted A -- B. The negations of , = and -- are denoted by ,
¢, ud /respectively. A set A is aproper subset ofB ifA = B but A /B.

PreBmimrie 
We will denote the empty et by . We will often denote a collection
(U.: •  A) of ets U. indexed by the set A simply by {U.} if explicit mention
of the index set is not necesutry. The union of the sets in the collection
{U.: A} will be denoted U U. or simply U U.. Similarly, their
intersection will be denoted  U. or simply  U..
U u. - {a: a belongs to mine U.}.
13 v. - {a: a belongs to every V.}.
The expression f: A --* B means that f is a mapping of the set A into the
et B. When describing a mapping by describing its eect on individual
elements, we use the special arrow -.; thus "the mapping m -,f(m) of A
into B" means thatfis a mapping ofthe set A into the set B taking the element
m of A into the elementf(m) of B. If U  A, then f] U denotes the restriction
off to U, and f(U) -- {b  B: f(a) -- b for ome a  U}. If C  B, then
f-l(C) -- {a  A:f(a)  C}. A mappingf is one-to-one (also denoted 1 : 1),
or injectie, if whenever a and b are distinct elements of A, thenf(a)  f(b).
A mapping f is onto, or surjectie, if f (A) -- B.
Iff: A .-* B and g: C .-* D, then the composition g of is the map
g of:f-(B  C) .-* D
defined by g of(a)-- g(f(a)) for every a f-(B  C). For notational
convenience, we shall not exclude the se in which f-(B  C)-- .
That is, given any two mappingrfand g, we shall.consider their composition
g ofns being defined, with the understanding that the domain ofg of may
well be the empty set.
The cartesia product A × B of two sets A and B is the set of all pairs
(a, b) of points a  A and b  B. If f: A - C and g: B - D, then the
cartesian product f × g of the maps f and g is the map (a, b)-* (f(a), g(b))
of A×BintoC×D.
We shall denote the identity map on any set by "id."
A diagram of maps such as
A
B
is called commutative ifg of.. h.
We shall always use the term function to mean a mapping into the real
numbers.

Let d  I be an Intcer, and let
Jf -- (a: a -- (as,..., a) where the a, are  numbers).
Then .' is the d.dimenslonal Euclidean space. In the case d ,= 1, we denote
the real line 1 simply by . The origin fO, ...  O) in Euclidean space of
any dimension will be denoted O. The noations [a,b] and (a,b) denote as
usual the intervals of the real line a  t  b and a  t  b respectively.
The function r,: IR' -- IR defined by
r  a) - a,,
where a *= (al,..., a,) ', is ed the h (¢nec)coordinatefinc:ian
on *. The canon/cal coordinate function r on  will be denoted simply
by r. Thus r(a) ,= a for each a  I. If f: Y-- I*, then we let
/, = g,.f,
wherej is ctned the/th component function off.
If f: I -- I and t  I, then we denote the derivative of fat t by
d ] df ] f(t + h) -- f(t)
If f: I"--* IR if 1 : i : n, and if t -- (tt,..., t,,)  IR", then we denote
the partial derivative off with re**pect to r at t by
Ifp  IR , then B,(v) will denote the open ball of radius • about p. The
open ball of radius • about the origin will be denoted simply by B(r). C(r)
will denote the open cube with sides of length 2r about the origin in I1 .
That is,
C(•) -- {(a,..., aJ  ': la,l < • for all i}.
We shall use C to denote the complex numbe• field and C" to denote
complex n-space,
C" -- {(zt .... , :,): :,  C for 1 < i < n}.
Unless we indicate otherwise, we shall always use the term neighborhood
in the sense of open neighborhood. If A is a subset of a topological space,
its closure will be denoted by AY. If p is a function on a topological space X,
the support of q is the subset of X defined by
supp ? -- q-t(ll -- {0}).
We use the Kroneckcr index
, i¢J.

Differentiable Manifolds
and
If o .= ( .... , ) is a d-tuple of non-negative integers, then we set
ar" at; "...
If • =. (0 ..... 0), then we let
DIFFEREIWT/ABLE OLDS
1.2 Definitions Let U c R be open, and letf." U--* IR. We say that
f is differentiable of class C  on U (or simply that f is CO), for k a non-
negative integer, if the partial derivatives BrTar" exist and are continuous on U
for [z]  k. In particular, f is C o iff is continuous. If f: U-- IR', thenf is
differentiable of class C A if each of the component functionsf = r of is C t.
We say thatfis C ® if it is C t for all k  O.
1.3 Definitions A locally Euclidean space M of dimension d is a
HausdorlT topological space M for which each point has a neighborhood
homeomorphic to an open subset of Euclidean space IR a. If q is a homeo-
morphism of a connected open set U c M onto an open subset of IR a,
? is called a coordinate map, the functions x == r o ? are called the coordinate
functions, and the pair (U, q0 (sometimes denoted by (U, xx ..... x)) is
called a coordinate system. A coordinate system (U, ?) is called a cubic
coordinate system if ?(U) is an open cube about the origin in IRa. If m  U
and q(m) ---- O, then the coordinate system is said to be centered at m.
1.4 Definitions A differentiable structure ." of class C t (1  k  oo)
on a locally Euclidean space M is a collection of coordinate systems
{(U,, q,): = ¢ A} satisfying the following three properties:
(a)
(b)
(c)
U U.=M.
% o q-x is C  for all =,/  A.
The collection r is maximal with respect to (b); that is, if
(U, q0 is a coordinate system such that q o q,-x and % o p-I are
CO for all a e ,4, then (U, q0 ¢r.

6
/f ,P" -- {(U,, ,?.): a eA} is any collection of coordinate systems
satisfying properties (a) and (b), then there is a unique difl'erentiable struc-
ture ," containing £r.. Namely, lct
£r =. {(U.?): ? • q.- and ?.o ?- Cforall
Then ," contains ,'e, dearly satisfies (a), and it is easily checked that ,"
satisfies (b) Now r is maximal by construction, and so ," is a dilerentiable
structure containing ,'.. Clearly ," is the unique such structure.
We mention two other fundamental types of difl'erentiable structures on
locally Euclidean spaces, types that we shall not treat in this text, namely.
the structure of class C" and the complex analytic structure. For a
differentable structure of clo C", one requires that the compositions in
(b) are locally given by convergent power series. For a complex amalytk
structure on a 2d-dimensional locally Euclidean space, one requires that the
coordinate systems have range in complex d-space C' and overlap holo-
morp    aly.
A d-dmeiomai dlfferentiable maold qcla C t (similady C" or complex
analytic) is a pair (M,,) cons/sting of a d-dimemional, second countable,
locally Euclidean space M together with a differentiable structure ," of
class C t. We shall usually denote the differentiable manifold (M,,') simply
by M, with the understanding that when we speak of the "differentiable
manifold M" we are considering the locally Euclidean space M with some
given differentiable structure ,'. Our attention will be restricted solely to
the case of class C °, so by differentiable we will always mean
of class CO. We also use the terminology smooth to indicate differentiability
of class CO. We often refer to differentiable manifolds simply as maifol,
with differentiability of class CO always implicitly assumed. A manifold
can be viewed as a triple consisting of an underlying point set. a second
countable locally Euclidean topology for this set, and a differentiable
stntcture. If X is a set, by a manifoldstrueture om Xwe shall mean a choice
of both a second countable locally Euclidean topology for X and a differ-
entiable structure.
Even though we shall restrict our attention to the CO case, many of our
theorems do, however, have C t versions for k < o, which are essentially
no more complicated than the ones we shall obtain. They simply require that
one keep track of degrees of differentiability, for differentiating a C t function
my only yield a function of class C t'-I if I  k <
Unless we indicate otherwise, we shall always use M and N to denote
differentiable manifolds, and M  will indicate that M is a manifold of
dimension d.
(a) The standard differentiable structure on Euclidean space I  is obtained
by taking ," to be the maximal collection (with respect to 1.4(b))
containing (1',0, where i: I'-- I' is the identity rap.

(b) t V  • te dimion!  vmr . en V h •
e du bi r e e orn fuon of • Oob orn
y on V. Such • global rn y uquy dn •
fftiabl. re  on V. s ffefiable su is d
ndt of e choi of basis, sin diffent b Ove C  overlaying
orate sysms. In fa, e chge of oinat is Oven simply
by a nsnt non-sinl ma.
(c) mpl n-s C" is a  mension vor space, d so,
by ample (b), h a natul sure  a mensional r
mfold.  (e,) is e onic mpl b  wch e, is e
n-tuple consisting of ros exit for a I in e  spot, 
is a I bis for C", d i dual bis is e no global
cina st on C'.
(d) e d-shere is e 
 n--(O ..... 0,1) and s--O,...,O,--l). en the sdard
ffenfiable sure on  is obn by ng  to be e ml
collation conng (- n, and (- s,), where , and 
e stergrapc projtions from n and s ively.
(e) An on subset U of a diffenfiable maold (M,) is i a
fferentiable ifold wi ffentiable
v - {(u.  u, .  u.  : (u., .
Units sifi oeise, on subs of differenfiable mifolds will
always be given is natur fferenfiable stature.
( e general linear grou Gi(n,) is e t of 1 n x n non-singular
r mais. If we identify in e obvious y e ints of ' wi
n x n real matces, en the denant bom a continues func-
tion on . Gi(n,) ives a manifold st  the on sut
of ' where e deteinant function ds not vani.
) Pruct noids. t (M ,) and (M ,  diffntiable
manifolds of dinsions d and d resfively. en M x M
becomes a differentiable manifold of mension d + d, wi differen-
fiable stu  e mim collation conning

1.6 Defln/t/ons Let U = M be open. We say that f: U--  is a
C" function on U(denoted f¢ C'(U)) iffo 0-' is C" for each coordinate
map  on M. A continuous map p: M-* N is aid to be dlfferentiable
of class C  (denoted p ¢ ®(M,N) or imly  ¢ C ) ifg o p is a C ® function
on ,p-(domain of g) for all C" functions g defined on open sets in N.
Equivalently, the continuous map V' is C ® if and only if  o V' ° '- is C ® for
each coordinate map - on M and  on N.
Clearly the composition of two differentiable maps is again differentiable.
Observe that a mapping V': M--* N is C ' if and only if for each m ¢ M there
exists an open neighborhood U of m h that V' ] U is C'.
THE SECOND AXIOM OF COUNTABILITY
The second axiom of countability has many comequences for manifolds.
Among them, manifolds are normal, metrizable, and paracompact. Para-
compactness implie the existence of partitions of unity, an extremely
useful tool for piecing together global functions and structures out of local
ones, and conversely for representing global structures as locally finite
uma of local one. After giving the neoessary definitions, we shall give a
imple direct proof of paracompactness for manifolds, and shall then derive
the existence of partitions of unity. It is evident that m_anifoids are regular
topological spaces and their normality follows easily from this and the
paracompactness.. We shall leave the proof that manifolds are normal as an
exercise. For the fact that manifolds are metrizable, see [13].
1.7 Deflnitiona A collection (U,) of subsets of M is a cover of a set
W Mif W [.JU s . It is an open cotr if each Us is open. A sub-
collection of the U s which still covers is called a subcover. A refinement
{Vt} of the cover {U.} is a cover such that for each/ there is an a such that
Vt  U,. A collection {A,} of subsets of M is locally finite if whenever
m ¢ M there exists a neighborhood W, of m such that Wm  A, #  for
only finitely many z. A topologimd spaoe is paracompact if every open cover
ha an open locally finite refinement.
1.8 Definition A partit/on of unity on M is a collection {?s: i ¢ I} of
C ® functions on M such that
(a) The collection of support {supp qs: i¢l} is locally finite.
(b) s qs(P) " I for all p ¢ M, and qs(P)  0 for all p ¢ M and i ¢ L
A partition of unity {qs: i  I} ia subordinate to the cover {U.: a ¢ A} if for
each i there exists an o such that supp q  U,. We say that it is subordinate
to the cover {Us: i¢I} with the same index set as the partition of unity if
supp q  Us for each i ¢ L

Second txiom of Coaabllit.v 9
1.9 Lemn Let X be a topological space which is locally compact
(each point has at least one compact neighborhood), Hausdorff, and second
countable (manifolds, for example). Then X is paracampact. In fact, each
open cover has a countable, locally finite refinement consisting of open sets
with compact closures.
rROOF We prove first that there exists a sequence {G,: i- I, 2,...}
of open sets such that
G, is compact,
(1) ¢' = ¢,+1,
x--UG,.
Let (U,: i- 1,2,...) be a countable basis of the topology of X
consisting of open sets with compact closures. Such a basis can be
obtained by starting with any countable basis and selecting the sub-
collection consisting of basic sets with compact closures. The fact
that X is Hansdorff and locally compact implies that this subcollection
is itself a basis. Now, let G -- U. Suppose that
Then letj be the smallest positive integer greater thanjk such that
and define
0 = U "
Th/s defines inductively a sequence {Gk} satisfying (1).

I0
comp and containod in the open set G+t -- G.. Fo ch i > 3
choo.e a linite .utver of e o c {U.  (- ,:
a ¢ A} of G -- G,_, d ch a  suv of e n 
{U.  O,: • e A} of  m  O,. s cl of  is
y n to  a counmb, iy  t of  on ver
{U, d s¢ of on  wi mt ci.
I.I0 Lemma There exists a non-negatlve C  function ? on ," which
equals 1 on the closed cube C(l'- and zero on the complement of the open cube
c().
eoo We need only let q be the product
O) e - (h • r,).. • (h • r,),
where k is a non.negative C '° function on the real fine which is I on
[--I,I] and zero outs/de of (--2,2). To construct such an h, we
start w/th the function
(2) f(t) - t S 0
which is non-negative, C ®, and positive for t > 0. Then the function
(3) e(t)- " f(0.
f(t) + f(1 - t)
is non-negative, C ®, and takes the value I for t : I and the value zero
for t : 0. We obtain the desired function k by setting
(4) h(t) -- e(t + 2)(2 - t).
I.II Theorem (Existence of Partltlons of Unity) Let M be a d/.0"er-
entiable manifold and (U.: • ¢ A) an open coer of M. Then there exists a
countable partition of unity (: i .= I, 2, 3 .... ) subordinate to the cover
 U.) with supp  compact for each i. Ifone does not require compact supports,
then there is a partition of unity {,} subordinate to the cover {U,} (that b,
supp ,  U) with at most countably many of the , not identically zero.
loov Let the sequence {G} cover M as in 1.9(1), and set Go -- .
For p ¢ M, let i, be the largest integer such that p ¢ M -- G,. Choose
an a, such that p ¢ U.,, and let (V,-) be a coordinate system centered
at p such that I/_.. U., r (G,,+l -- G,'- and such that -(V) conUdns
the closed cube C(2). Define
(1) P -- elsewhere

Tangent Vectors and Differentials 11
where ? is the func.on 1.10(1). Then p, is a C  function on M which
has the value l on some open neighborhood W, ofp, and has compact
support lying in V  Ua, C (O,+s" ,,). For each i : 1, choose a
finite set of points p in M whose corresponding W neighborhoods
cover O- -- O-t. Order the corresponding V functions in a luence
,p,j -- 1, 2, 3 ..... The supports of the *p form a locally finite family
of subsets of M. Thus the function
(2) • - Y
is a weli-defined C  function on M, and moreover (p)  0 for each
pM. Forenchi--l,2,3,...define
(3) -
Then the functions {?: i--1, 2, 3,...} form a partition of unity
subordinate to the cover {Ua} .with supp ? compact for each L If we
let ?, be identically zero if no % has support in U,, and otherwise let
¢p, be the sum of the ¢p with support in U,, then {¢Pa} is a partition of
unity aubordinate to the cover {U,} with at most countably many of the
?, not identically zero. To see that the support of ?, lies in U,, ob-
serve that if. is a lcally finite family ofclosed sets, then U A -- i.J A.
Observe, however, that the support of ?, is not necessarily compact.
Corollary Let 0 be open in M, and let A be closed in M, ,,ith A  G.
Then there exists a C ® function  : M -  such that
(a) o< <
(c) supp ?  O.
PaOOF There is a partition of unity (,} subordinate to the cover
(G,M--A} of MwithsuppG and suppM--A. Then ?
is the desired function.
TANGENT VECTORS AND DIFFE
1.12 A vector v with components vt , .... v at a point p in Euclidean
space  can be thought of as an operator on differentiable functions.
Specifically, iffis differentiable on a neighborhood ofp, then v assigns to
fthe real number v(f) which is the directional derivative off in the direction
v at p. That is,
1 r I

This operation of tbe vtor  on diffrentiabk functions satisfies two
important properties,
(2) g) - +
whenever f and g are differentiable near p, and t is a real number. The
first property says that v acts llnearly on functions, and the second say that
v is a derivation. This motivates oar defm/tion oftangent vectors on manifolds.
They will be directional derivatives, that is, linear derivations on functi6ns.
The operation of taking derivatives depends only on local properties of
functions, proprt/es in arbitrarily small neighborhoods of the point at
which the derivative is being taken. In order to express most conveniently
this dependence of the derivative on the local nature of functions, we intro-
duce the notion of germs of functions.
1.13 DeAlnltiom Let m • M. Functions f and g defined on open sets
containing m are said to have the same germ at m if they agree on some
neighborhood of m. This introduces an equivalence relation on the C ®
functions defined on neishborhoode of m, two functions being equivalent
if and only if they have the same germ. The e_quivalene classes are called
germs, and we denote the set of germs at m by '. ,. Iff is a C  function on
a neighborhood of m, then f will denote its germ. The operations of addition,
scalar multiplication, and multiplication of functions induce on ., the
structure of an algebra over I. A germ f has a well-defined value f(m) at m,
namely, the value at m of any representative of the germ. Let F,,  ,, be
the set of germs which vanish at m. Then F,, is an ideal in ,,, and we let
F denote its kth power. F is the ideal of ,, consisting of all finite linear
combinations of k-fold products of elements ofF,,. These form a descending
sequence of ideals ,, = F,, = F,, a = F, s = ..-.
1.14 Definition A tangent vector v at the point m  M is a linear deriva-
tion of the algebra ,,. That is, for all f, I  m and 
(a) v(f + -- +
(b) v(f" |) == f(m)v(g) ÷ |(m)v(f).
M,, denotes the set of tangent vectors to M at m and is called the tangent
space to M at m. Observe that if we define (v + w)(f) and (v)(f) by
(v + Xr) - vCr) +
(l)
-
whenever v. w  M,, and  s I. then v + w and v again are tangent vectors
at m. So in this way M. becomes a real vector space. The fundamental
property of the vector space M,. which we shall establish in 1.17. is that
its dimension equals the dimension of M. This definition of tangent vector

Tangent Vectors and Differentials 13
is not suitable in the C  case for 1  k  a. (We will discuss the C  case
• further in 1.21.1 Wc give this definition of tangent vector for several reasons.
One reason is that it is intrinsic; that is, it does not depend on coordinate
systems. Another reason is that it generalizes naturally to higher order
tangent vectors, as wc shall see in 1.26.
1.15 Ife is the germ of a function with the constant value c on a neighbor-
hood of m, and if v is a tangcnt vector at m, then v(e) , 0, for
v(c) .- cv(1),
and
v(1) ffi v(1. I) ffi Iv(1) + Iv(1) - (I).
1.16 Lemma M, is naturally isomorphic with (F,/F)*. (The symbol *
denotes dual vector space.)
],aOOF If v C M.,, then v is a linear function on F, vanishing on F,, t
because of the derivation property. Conversely, if #c (F,]Fmt) *, we
define a tangent vector v¢ at m by setting vf(f) z '({f -- f(m)}) for f c P,.
(Here f(m) denotes the germ of the function with the constant value
f(m), and { } is used to denote cosets in F,FS.) Linearity of v¢ on P,
is clear. It is a derivation since
v¢(f. g) z f({fg -- f(m)g(m)))
- (((f - f(.))(g - g(.)) + (-)(g -
"t" g(m)'((f -- f(m)))
ffi f(m)v¢(g) -t- g(m)v(f).
Thus we obtain mappings of M, into (F,,,/F,,,m) *, and vice versa. It
is easily checked that these are inverses of each other and thus are
iso morphisms.
1.17 Theorem dim (F,F,*) ffi dim M.
The proof is based on the following calculus lemma [31].
Lemma If g is of class C  (k  2) on a convex open set U about p in .,
then for each q  U,
- or I
+ y (,.,(q/- r,(,))(,',(q) - ,',(p)) (1 - t/---- dr.
.: l' r I (S-f(--))
in ,vartcular, if   C ®, then the second mmation in (1) determines an
element of F  since the integral a a function of  is of class C ®.

14
lqOOF ot 1.17 Let (U,?) be a coordinate system about m with co-
ordinate funct/ons x1,..., xd (d -- dim M). Let f e Fro. Apply
(I) to/'. q-, and ¢ompo with q to obtn
on a neighborhood of m, wber h  C °. Thus
z ,-1  (fO)r ¢p-t)]¢¢,(xt _ xt(m)) mod Fms.
f
Hence ((x, - (,-)):  = l,...,d} ,pan, F./F.*. Consequently
dim FmFm t  d. We claim that these elements are linearly independent.
For suppose that
Thus
 o,( - (=)) c F:.
•
But this implies that
forj = 1, .... d, which implies that the a, must all be zero.
Corollary dim M, -= dim M.
lag In practice we will treat tangent vectors as operating on functions
rather than on their germs. Iff is a differentiable function defined on a
neighborhood of m, and v ¢ M,,, we define
(1) v(f) - v(f).
Thus v(f)=, v(g) whenever f and g agree on a neighborhood of m, and
clearly
v(f + ) = v(J') + (g) ( c ),
(2)
v(f" g) --f(m)v(g) "b g(m)v(f),
where f+ .g and f.g are defined on the intersection of the domains of
definition of f and g.
1.19 Defln/tion Let (U,?) be a coordinate system with coordinate
functions xt .... , xd, and let m  U. For each re (1,..., d), we define

Tangent Vectors and Differentials
a tangent vector (OlOx,)]. ¢ M. by setting
(1)
for h funcfionfwhich ia & on a nO of re. We inret (1)
 the on dvative of fat m in e
u e
(2) x, I=
(c)
Renutrks on IA9
Clearly ((a/ax)l,.)(f) depends only on the germ of fat m, and (a) and
(b) of 1.14 are satisfied; so (a/Ox)l,. is a tangent vector at m. Moreover,
{(O]Ox,)l.: i z 1 ..... d} is a basis ofM.. Indeed, it is the basis
dual to the basis {{x, -- xt(m)}: i ,= 1 ..... d} ofF,F,.  since
x ,,,(x - x(m))- ..
If v  Mm, then
Simply check that both sides give the same results when applied to the
functions (xs -- x(m)).
Suppose that (U,9) and (V,p) are coordinate systems about m,
with coordinate functions xx ..... x andyx, .... y respectively. Then
it follows from remark (b) that
Observe that (/x) depends on q and not only on x. In particular,
if xl were equal to yl, it would not necessarily follow that O/Ox equals
O/ay .

(d) If we apply Definition 1.19 to the canonical coordinate system
rt ,. • •, rd on IR d, then the tangent vectors which we obtain are none
other than the ordinary partial derivative operators (/r).
1.21 Our proof of the finite dimensionality of FF,,* certainly fails in the
C t case for k < oo ince the remainder term in the lemma of 1.17 will not
be a sum of product of C t functions, and the lemma doesn't even make
sense in the C t case. In fact, it turns out (see [21]) that F,F, t is alway
infinite dimensional in the C  case for 1  k < oo. There are various ways
to define tangent vectors in the C t case in order that dim M., z dim M
(all of which work in the C ® case, too). One way is to define a tangent vector
v at m as a mapping which assigns to each function (defined and dilTerentiable
of class C  on a neighborhood of m) a real number v(f) such that if (U,q)
is a coordinate system on a neighborhood of m, then there exista a list of
real numbers (at,..., a) (depending on q) such that
Then the space M. of tangent vectors again turns out to be finite dimensional,
with a
1.22 The Differential Let p: M-- N be C ®, and let m ¢ M. The
differential of  at m is e liar map
n  foHo. If v e M.,  (v) i to  a tangent vtor at (m),
so we d how it ot on funcons. Let g  a  function on a
neighboo of (m). Define d(v)) by tting
(2) (v))
It i, ily chk at  is a liner map of M. into N.. Stctly
sng, s map should  denot   M., or sim:y d.. However, we
omit the subpt m when ere
is cl non-silar at m if @. is non-singul, at is, ife kd of (I)
cois of 0 alone. e dual
is defi  usual by uing at
(4) ()() = (())
whenr f¢. d vfM..  e si c of a C  funcon
fi M  , if v f M. and f(m) = re, then
(

Tangent Vectors and Differentials 17
In this ease, we usually take dfto mean the clement of M* defined by
(6) df(v) = v(f).
That is, we identify dfwith f(¢o), where ¢0 is the basis of the 1-dimensional
space IR* dual to (d/dr)],o. Particular usage will be clear from the context.
1.25 Renutrks on 1.22
(a) Let (U, xt,..., x) and (V, yt ..... y¢) be coordinate systems about
m and p(m) respectively. Then it follows from 1 2(2) and 1.20(b) that
The matrix {8(y, o )/Sx} is called the Jacobian of the map V' (with
respect to the given coordinate system). For maps between Euclidean
spaces, the Jacobian will always be taken with respect to the canonical
coordinate systems.
(b) If (U, xt ..... x) is a coordinate system on M, and m c U, then
{dx,],} is the basis of M* dual to {/x],}. If f: M-- IR is a C ®
function, then
df, ,-tx, .
(c) Chain Rule. Let p: M-Nand q: N-Xbe C ® maps. Then
d(+ o v)= - d+,=, o
or simply d( o ) ffi d o d. It is a useful exercise to check the form
that this equation takes when the maps are expressed in terms of the
matrices obtained by choosing coordinate systems.
(d) If : M - N and f: N- I are C ®, then (df,) - d(fo )m, for
(df.)(v) ffi df(d(v)) ffi d(f o ).(v) whenever v c M..
(e) A C ® mapping : (a, b)- M is called a smooth curve in M. Let
t  (a, b). Then the tangent vector to the curve ¢ at t is the vector
We shall denote the tangent vector to  at t by O(t).
a t

Now, if v is any dement of M,, then v is the tanscut vector to a
smooth curve in M. For one can s/reply choose a coordinate system
(U,?), centered at m, for which
Then v is the tangent vector at 0 to the curve t , ?-z(t, 0,..,, 0).
One should observe that many curves can have the same tangent
vector, and that two smooth curves ¢ and " in M for which ¢(t,) .=
v(t.) -- m have the same tangent vector at t. if and only if
dr I,. dr .
for all functiomfwhich are C ® on a neighborhood ofm.
If ¢ happens to be a curve in the Euclidean space I1-, then
- +... + ,,;
dr , ,(,) ,
If we identify this tangent vector with the element
of I1", then we have
r h ' dr IJ
0(0 -= lira (t + h) --
Thus with this identification our notion of tangent vector coincides,
in this special case, with the geometric notion of a tangent to a curve
in Euclidean space.
1.24 Theorem Let / be a C ® mapping of the connected manifold M
into the manifold N. Suppose that for each m ¢ M, d,p,, -, O. Then  is a
constant nmp.
PROOt Let n ¢,/,(M). ,e--l(n) is closed. We need only show that
it is open. For this, let m ¢--1(n). Choose coordinate systems
(U, x, .... x) and (V, yt .... ,Ye) about m and n respectively, so
that ,(U) = F. Then on U,

Thus the ftmctionsy, • *p are constant on U. This implies that ,p(U) -- n;
hence ,p-t(n) is open and onsequently ,p-(n) -- M.
We shall now see that in a natural way the ollection of all tangent
vectors to a differentiable manifold itself forms a differentiable manifold
called the tangent bundle. We have a similar dual object clled the cotangent
bundle formed from the linear functionals on the tangent spaces.
1.25 Tangent and Cotangent Bundles Let
with differentiable stntcture ,'. Let
T(M)- U M,,
(1) T*(M) -- U M*..
M be a C  manifold
There are natural projections:
.: T(M)-- M, r(v)--m if v e M.,
(2)
w*: T*(M) "- M, w*(') " mff " ¢ M*m.
Let (U,q) .$" with coordinate functions x,... ,x,. Define : ,r-(U) --
u and *: (,*)-(U)-, u by
{v) = (x,(-{v)) .... , x,(.{)), d,) .... , d))
for all v  vr-*(U) and •  (*)-ffi(U). Note that  and * are both one-to-one
maps onto open subsets of t. The following steps outline the construction
of a topology and a differentiable structure on T(M). The construction for
T*(M) goes similarly. The proofs are left as exercises.
(a) If (U,) and (V,?) e,r, then  o -* is C ®.
(b) The collection {-*(W): W open in IR t, (U,)} forms a
basis for a topology on T(M) which makes T(M) into a 2d-
dimensional, second countable, locally Euclidean space.
(c) Let ' be the maximal collection, with respect to 1.4(b), containing
{(-*(u),): (u,v) e,r).
Then ' is a differentiable structure on T(M).
T(M) and T*(M) with these differentiable structures are called respectively
the tangent bundle and the cotangent bundle. It will sometimes be convenient
to write the points of T(M) as pairs (re,v) where m  M and v  M, (and
similarly for T*(M)).

whe dm,v)- dp,(v) whenev v  M,. It is y k t (4)
ba p.
I TtV h is 
w lk at ,  (FF)*, for s nt of  o  ia
gon W  o t . W  f a moment
R t = is e b or g of fcons at m. F= c  is
• e id o ge ng at m, d F @  int  I) is e id o
ofF,.
e or s F  is   s of   dertl at m,
d we no it by 'M..  fo f do e ge off at m, d ( )
 no  in F . f a enble funon on a nr-
h of m. We e e  o dert f off at m by
A  or t£ent ¢tor at m is a   fu on P. vn s
on  d vng  on   of gs of fom t on a
iborh of m. e  fi s of  o t vton at
m   do by M. We  a m i of M. * wi
(*M,)"  y  or t vor  W F.  a n
funcon on F. vhi on , d he yiel  element of (*M";
d convly  elt of (M,)" uuely  a  fuon
 F= i on +:, a s n uquely  a  o ngent
or by d it W   of t fuons.
We  fie up  noon of   t vor wi e us
nion of  or vve  Eucd spa by lng at e fos
t  gent vt d ffs e in a cte sy.
 (U,)  a na s aut m wi na funcons
x,..., x s t ( is a nv o  in Euc s .
t  = (, ....   a fi of noon,five in. In adtion to
our no of I. I, we I
- - - -
f a  fon on U.  it fows fm e le of 1.17, t
(2) f--/(m)+ aa(x--m))'+  haCx--m),
 IInoteewity
 t I  ui.

Tangent Vectors and Dlfferenrlala 21
where the h, are C ® functions on U and where
O)
Hence
(4)
Thus the collection
I (3"o -)[
a.- .. at" ,,./
[{(x - s(m))'}: I < [1 < k]
spans tM,,. The proof that thcse clcmcnts arc lincarly independent in
tMM is thc obvious generalization of the proof for thc case k -= 2 which was
treated in 1.17. Thus thc collection (5) forms a basis oftMM. Consequcntly
M is finite dimensional with dimension equal to the binomial coefficient
 • . As the dual space of Ms , Ms is also finite dimensional
with the same dimension. Since M is identified with (M)*, and these
spaces are finite dimensional, we have a canonical isomorphism of M, with
(M)*, under which the element of d'e tMM, considered as an element
of (M)*, satisfies
(6) #y(v) = v(r).
Let
(7) ax" ar"
Since the derivative is linear, and since the value of 8"./78 at m depends
only on the germ off at m and vanishes iff is constant on a neighborhood
of m or if f is an [] + 1-fold product of functions which vanish at m, then
(8/8x)1, is an []th order tangent vector at m. It follows that
is the basis of M dual to the basis (5) of tM. If  is a/cth order tangent
vector at m, then
v=Yb.
(9)
where
(10) b, = (l((x -- x(m))').
In terms of the basis ($), equation (3) becomes
(11) a. =  8x-"  ,.

As in the case of first order tangent vectors,  customarily think of
tangent vectors as operating on the functions themselves rather than their
whmmverf is C ® on a neighborhood of m and v is a tangent vector of any
order at m.
Finally, just as there are natural mappings of tangent vectors ud diffm-
rials assoiatl with a diffmentiable mp?: M;, so are there linear
mappings
(13)
dby
(14) (,). d.( o )
whener v e M  g is a & fuon on a Obo of (m).
It is y ch t (14)  in ne  mppi (13) d t
• e mapn dt and  m du.
r fion of a t or mnnt vtor in  fion  
Dfion 1.14 in  of 1.16. Mo, we   
infiom of e t o ffnfi df of a function f; e in-
prfion (13) a wi our o0 dfion 1.(1), e inmtion
(6)   1.(6), d  have e adon inaction (1).
SUBMANOLD, DIFFEOMORPHISMS, AND THE
INVERSE FUNCTION THEOREM
(a)
(c)
Definitions Let p: M -- N be C ®.
p is an immersion if dlo, is non-singular for each m ¢ M.
The pair (M,p) is a submanifold of N if V is a one-to-one immersion.
p is an imbedding if p is a one-to-one immersion which is also a
homeomorphism into; that is, p is open as a map into ,p(M) with
the relative topology.
p is a diffeomorphism if p maps M one-to-one onto N and ,p--I
is C °o .
1.28 Remarks on 1.27 One an, for example, immerse the real
line I into the plane, as illustrated in the following figure, so that the first
case is an immersion which is not a submanifold, the second is a submanifold
which is not an imbedding, and the third is an imbedding.

Submanlfolds, Dffeomorph'n, Inrse Function Theorem 23
Immersion,
but not • Submanifold
;ubmanifold,
but not an Imbeddins
Imbedding
Obve that if (U,) is a coordinate sstem, then ?: U-- ?(U) is a
diffeomorphism.
The composition of diffeomorphisms is again a diffeomorphism. Thus
the relation of being diffeomorphic is an equivalence relation on the collec-
tion of differentiable manifolds. It is quite possible for a locally Euclidean
space to possess distinct differentiable structures which are diffeomorphic.
(See Exercise 2.) In a remarkable paper, Milnor showed the existence of
locally Euclidean spaces (S  is an example) which possess non-diffeomorphic
differentiable structures [19]. There are also locally Euclidean spaces which
possess no differentiable structures at all [14].
If p is a diteomorphism, then d, is an isomorphism since both
(d o d-l)l¢, and (d -1 o d)[, are the identity transformations. The
inverse function theorem gives us a local converse of this---whenever d,
is an isomorphism,  is a dif[eomorphism on a neighborhood of m. Before
we recall the precise statement of the inverse function theorem, we give a
definition which will be needed in the corollaries.
1.29 Definition A set y ..... y of C ® functions defined on some
neighborhood of m in M is called an independent set at m if the differentials
dye,..., dy form an independent set in M.
1.30 Inverse Function Theorem Let U   be open, and let
f: U--  be C ®. If the Jacobian matrix
r .- ..... 
is non.singular at re  U, then there exists an open set V with re  V  U
such tt f I Vmaps v one-to-one onto the open serf(V), and (fl V)-a is C .
This is one of the results we shall assume from advanced calculus. For
a proof, we refer the reader, for example, to [31] or [6].

Corollary () As.non that : M-+ N b C ®, that m • M,  t
: M,  Nt,j  an rp.  tm  a igr U of m
 tt : U  (U)  a d oo t on t (U)  N.
• aF Oe at  M-- m N, y d.   
(V,9) aut m d (W, aut (m) wire   W. t 9(m) --p
d m))- q. e ffnfi 0f  map  o   -x[ ( is
non-n atp.    function  yi a diff
monism g:  0) on a neirh 0 of p  0 = (.
en  o  o 9 is   ffmosm on  ngrfio
U -- p-( of re.
Corollary (b) Suppose that dim M I d and that Yx,... ,Y, i an
independent set of functions at me ¢ M. Then the functions Yx , . . . , Y form
a coordinate system on a neighborhood of me.
PROOF Suppose that the y are defined on the open set U containing
me. Define ,p: U--, IIa by
- (y,(m),..., y,(m)) (m v).
Then ,p is C ®. Now p is an isomorphism on (11=.))* since
which implies that ),p],.) takes a basis to a basis. Consequently, the
differential d,. (which is the dual of ),p],.)) is an isomorphism. So
the inverse function theorem implies that ,p is a diffeomorphism on a
neighborhood V c U of me, and consequently the functions Ys .... , Y,
yield a coordinate system when restricted to V.
Corollary (c) Suppose that dim M  d and that Yx, . . . , Ys, with i < d,
is an hulependent set of functions at m. Then the 7 form part of a coordinate
system on a neighborhood of m.

Submantfolds, Diffeomorphir, Inverse Function 7orem 
lmoor Let (U, xl ..... x,) be a coordinate jtem about m. Then
(dr, .... , dye, dxl,..., dx,) spans M s . Choose d-- I of the
so that (dy ..... dye, dx, 1 .... , dxe_,) is a basis of M. Then apply
Corollary (b).
Corollary (d) Let o: M -- N be C ®, and assume that do: M, -- N,
surjective. Let x. .... , x form a coordinate system on some neighborhood
of o(m). Then x o o ..... x o o form part of a coordinate system on some
neighborhood of m.
Poor The fact that dm is surjective implies that the dual map
O]rm is injecfive. Thus the functions (xo: i 1 .... ,1 are
independent at m since 6o(dx)  d(x o o). The claim now follows from
Corollary (c).
Corollary (e) Suppose that y. .... ,y is a set of C ® functions on a
neighborhood of m such that their differentials span l . Then a bset of the
y forms a coordinate system on a neighborhood of m.
lmoor Simply choose a subset whose differentials form a basis of
MS, and apply Corollary (b).
Corollary(f) Let o: I-¢ N be C, and assume that d°: Mm- Nm is
injective. Let xx .... , xt form a coordinate system on a neighborhood of
o(m). Then a subset of the functions (x, o o forms a coordinate system on
neighborhood of m. In particular, o is one-to-one on a neighborhood of m.
Paoor The fact that d, is injective implies that ¢o]r, is surjecfive.
This implies that (d(x,oo) Oo(dx,): i----1,... ,k spans
This corollary then follows from Corollary (e).
1.31 The situation often arises that one has a C ® mapping, say , of a
manifold N into a manifold M factoring through a submanifold (P, cp) of
M. That is, (N)  (P), whence there is a uniquely defined mapping
of Ninto P such that ¢p o s ---- . The problem is: When is s of class C®?
This is certainly not always the case. As an example, let N and P both be the
real line, and let M be the plane. Let (,) and(,) both be figure-8
submanifolds with precisely the same image sets, but with the difference that
as t - 4- oo, (t) approaches the intersection along the horizontal direction,
but (t) approaches along the vertical. Suppose also that (0)  ¢p(0) - 0.
Then o is not even continuous since o-(--1,1) consists of the origin plus
two open sets of the form (,-/- oo), (--oo,--) for some •  0.

1.32 Theorem SuFo that : N- M Is C , that (P,)  a oM
¢ti, t  a  pin g  of N to P .ch tt  •  -- .
N  M 
o p
Anoer imt  in w 0 ia nfinuom u wh (,) h
 in maold of  involufive bufion on M, m we   in
1.62.
prove t it i &. It u  how at P    by 
o ym (U,) ch t the map r    to e o
pmon  of  Ohm a b  (ob by ng n
of e ci om  m 0) h   map r --   Y  
whi is &.
1.33 Further Remarks on Submanifolds Submanifolds (-'VI,) and
(Nt, q,) of M will be called equivalent if there exists a difreomorphism
o(: NI -'* N: such that
This is an equivalence relation on the collection of all subman/folds of M.
Each equivalence class  has a unique representative of the form (A,i)
where A is a subset of M with a manifold structure such that the inclusion
map t: A -- M is a C ® immersion. Namely, if (N,?) is any representative

bmanlfolds, Dtffeomorphns, Inrse Function Theorem 27
of , then the subset A of M must be q(N). We induce a manifold structure
on A by requiring q: N--* A to be a ditfeomorphism. With this manifold
structure, (A,i) is a submanifold of M'equivalent to (N,q). This is the only
manifold structure on A with the property that (A,t) is equivalent to (N,q);
thus this is the unique such representative of .
The conclusion of some theorems in the following sections state that there
exist unique submanifolds satisfying certain conditions. Uniqueness means
up to equivalence as defined above..In particular, if the submanifolds of
M are viewed as subsets A = M with manifold structures for which the
inclusion maps are C ® immersions, then uniqueness means unique subset
with unique second countable locally Euclidean topology and unique
differentiable structure.
In thecase of a submanifold (A,t) of M' where t is the inclusion map,
we shall often drop the t and simply speak of the submanifold A" = M.
Let A be a subset of M. Then generally there is not a unique manifold
structure on A such that (A,t) is a submanifold of M, if there is one at all.
For example, the diagrams in 1.31 illustrate two distinct manifold structure
on the figure-8 in the plane, each of which makes the figure-8 together with
the inclusion map a submanifold of IIs. However, we have the following
two uniqueness theorems which involve conditions on the topology on 4.
(a) Let M be a differentiable manifold and 1 a subset of M. Fix a topology
on A. Then there is at most one differentiable structure on A such that
(A,i) is a submanifold of M, where i is the inclusion map.
(b) Again let A be a subset of M. If in the relative topology, A has a differen-
tiable structure such that (A,i) is a submanifold of , then A has a
unique manifold structure (that is, unique second countable locally Euclidean
topology together with a unique differentiable structure) such that (A,i)
is a submanifold of M.
We leave these to the reader as exercises. Result (a) follows from an applica-
tion of Theorem 1.32. Result (b) depends strongly on our assumption that
manifolds are second countable, and for its proof you will need to use the
proposition in Exercise 6 in addition to Theorem 1.32.
1.4 Slices Suppose that (U,q) is a coordinate system on M' with
coordinate functions xz ..... x,, and that c is an integer, 0  c  d. Let
a  9(U), and let
(I) s = u: = i = c + I .... , a).
The subspace S of M together with the coordinate system
(2) Is:/= I,..., c)
forms a manifold which is a submanifo]d of M called a slice of the coordinate
system

1.35 Proposition let q,: M e- N a be an immersion, and let m  M.
Tben tbere exists a cubic.centered coordinate system (g, cp) about p(m) and a
neighborhood U of m such that  [ U ts 1:1 qnd U) is a slice of (v,).
l,gool Let (W,) be a centered coord/nate system about 0(m) with
coordinate functions yt,... ,y. By Corollary (f) of 1.30 we can
renumber the coord/nate functions so that
(l) .r,oo
is a coordinate map on a neighborhood V' of m where r.: * - "
is projtion on the fn c ordinatm.  tmctions {x,} on
yf, (t = ,..., c)
(2) x, -,
,--y,opo-'or,*,. (i--c+ l,...,d).
The functions {x,} are independent at p(m), sinc at p(m),
dy (i -- 1,..., c)
(3) dx, - [dy, + .,, dy, (i -- c + 1 .... , )
for some constants ao. By Corollary (b) of 1.30 the {x,} form a coordinate
system on a neighborhood of p(m). Let F b a neighborhood of p(m)
on which the xl, .... x, form a cubic coordinate system. Denote the
corresponding coordinate mp by . Let U--,p-l(V) V'. Thn
U and (V, qO are the required neighborhood and coordinate system.

Subnumol, Deomorphl,s'r, Inoerse Function eorem 2
We emphasize that this proposition only ays that there is a neighborkood
U of m such that ,p(U) is a slice of the ccordinate system (V,). Even if
(M,p) is a submanifold of N, it my well be [lint p(M) r Ix is fr from being
a slice or ven a uion of slices. For an example, consider gain the figure8
ubmanifold of the plane:
However, in the case that (M,p) is an imbedded submanifold, the coordinate
system (V, q0 can be chosen so that all of p(M) f V is a single slice of V.
Let us now consider the question of the extent to which the set of C ®
functions on a manifold determines the set of C ® functions on a submanifold.
Let (M,0) be a submafiifold of N. Then, of course, if f C(N), then
fl M is a ¢® function on M. (More precisely,fo p is a C ® function on .M.)
In general, however, the converse does not hold; that is, not all C ® functions
on M arise as the restrictions to M of C ® functions on N. For the converse
to hold, it is necessary and sufficient to assume that p is an imbedding and
that p(M) is closed. We prove the sufficiency in the following proposition,
and leave the necessity as Exercise 11 below.
1.36 Proposition Let p: M-*N be an imbedding such that p(M)
is closed in N. If g  C(M), then there exists f  C®(N) such that f o p I g.
To simplify notation, we shall suppress the map 0 and consider M c N.
moov For each pointp  M there exists an open set O, in N containing
p and an extension of g from O, f M to a C ® function , on
One simply has to take O, to be a cubic-centered coordinate neighbor-
hood ofp for which M f O, is a single slice, and then define , to be the
composition of the natural projection of O, onto the slice followed by
g. The collection {O,: p  M} together with N- M forms an open
cover of N. By Theorem 1.11, there exists a partition of unity {q}, with
j  1, 2 ..... subordinate to this cover. Take the subsequcnce (which
we shall continue to denote by {%}) such that supp % f M # Z. For
each such j, we can choose a point p such that supp % c O,. Then
f  p,, is a C ® function on N, andfl M  g.

IlVLIr FUNCTION THEOI.I
From the invm function thoorem we shall obtain two thorm which will
provide us with an extremciy useful way of.prov/n$ that cm/n subsets of
manifolds are subman/folds. Under su/table cond/tiom on a different/able
map, the inverse image of a subman/fold of its ranp will be a submsnifold
of its doma/n. We first recall the statement of the clau/cal implicit function
theorem. Th/s is s/reply a local (but somewhat more expl/cit) version of the
first "implicit funct/on" thoorem (1.38) that we shall prove for man/folds.
We suggest that the reader supply a proof of 1.37 after read/ng 1.38.
1.3"/ Implicit Ftmcflon Theorem Let U  1o-4 x I' be open, and
let f: U-  be C ®. We denote the canonical coordinate system on
- g a by (rx, .... r.4 , $, . . . , sa). SWposethatat tbepoint (r I ,) c U
and that the matrix
f (r ,ao) -- O,
is non-singular. Ten tere exists an ogt netghbord g of re in :'-4 and
an open etghborod Ff of $ in  auch that V x W = U, and tr¢ exiata
a C" map g: V-- Wuch that for each (p]) v V g W
(1) f(p,q) -- 0  q . g(p).
W

Implicit Function Theorems 1
1.$8 Theorem Assume that  : M¢'- N is C, that n is a p°int °f N'
that P I p-t(n) is non.empty, and that do: M, -- N, is surjective for all
m  P. Then P has a uniq manifoM structure such that (P,t) is a submanifold
of M, where i is the inclusion map. Moreover, i: P-- M is actually an
imbedding, and the dimension of P is c -- d.
rgoov According to result (b) of 1.33, it is sufficient to prove that in
the relative topology, P has a differentiable structure such that (P,i) is
a submanifold of M of dimension c -- d. For this it is sufficient to prove
that if m v P, then there exists a coordinate system on a neighborhood
U of m in M for which P  U is a single slice of the correct dimension.
Let xl,. • •, x be a coordinate system centered at n in N. Then since
d: M,-- N, is surjecfive, it follows from Corollary (d) of 1.30 that
the collection of functions
,  x, o o: i  1,... , d},
forms part of a coordinate system about m  M. Complete to a co-
ordinate system yl, • .., y, Y .... , y on a neighborhood U of m.
Then P f U is precisely the slice of this coordinate system given by
In this theorem the inverse image of a point is shown to be a submanifold
as long as the differential is surjective at each point of the inverse image.
A point can be thought of as a 0.dimensional submanifold. We now generalize
this theorem by proving that under suitable conditions the inverse images
of higher dimensional submanifolds are themselves submanifolds.
1.$9 Theorem Assume that o: M-N  is C ® and that (0%) is a
submanifold of N. Suppose that whenever m  -x(p(O)), then
(1) N.  dv(M.) + d(O_l.)
(not necessarily a direct sum). Then if P ffi -Z(q(O)) and is non-empty,
P can be given a manifold structure so that (P,i) is a submanifold of M, where
i is the inclusion map, with
(2) dim M -- dim P ---- dim N -- dim O.
Moreover, if (0,) is an imbedded submanifold, then so is (P,i), and in this
ease there is a unique manifoM structure on P such that (P,i) is a submanifold
of M.
In general, if (O,) is not an imbedding, P need not have the relative
topology, and there is no unique manifold structure on P such that (P,i)
is a submanifold. We leave it to the reader to supply examples.

32 MaIfeIds
moo.F The proof will consist of locally reducing this cue to the case
of Theorem 1.38. Let p ¢ O. By Proposition 1.35, we can pick a
neighborhood W of p and a centered coordinate system (V,.) with
coordinate functions xx,..., x, about )tlch that q(W) is the slice
(3)
Let
(+)
(5)
Let
(6)
and let
,p, -- +.to ,,r o p I U: U- A *-e.
Now, (3) through (7) imply that
(8)
For each point m  V,I-(0), dv, [ M, is surjective since d(r
is surjective and since by (1), (5), and (7)
d('." o 'r)(N,,,,<.,+) -- d,(M.) -I- {0}.
Thus by 1.38, *p-'(?(W)) has a unique manifold structure such that
(p-'(?(W)), 0 is a submanifold of M; and in this manifold structure,

Implicit Function Theorems 33
-((W)) has the relative topology and has dimension equal to
dim M -- dim N -F dim O. Cover O by a countable collection
{W,: i == 1,2, 3,...) of such sets. Then
If/#j, the submanifolds o-l(q(W,)) and -l(q(W)) intersect in open
subsets of each other. It follows that the union of the topologies on the
various -(q(W,)) forms a basis for a topology on P. With this topology,
P is a locally Eucfidean space with dimension equal to dim M-
dim N+ dim O; and, moreover, P is second countable since (9)
expresses P as a countable union of second countable open subsets.
The manifold structures on the various -(q(W,)) are compatible on
overlaps because of the uniqueness of the manifold structure on
-s(q(W,)) (or any open subset of-s(q(W,))) such that
is a submanifold of M (by result (b) of 1.33). Thus the collection of
coordinate systems on P containing the coordinate systems on the
various -l(q(W,))and maximal with respect to 1.4(b) forms a differen-
tiable structure on P. That (P,i) is a submanifold of M now follows
immediately from the fact that the (-:(q(W,)),i) are submanifolds.
If (O,q) is an imbedding, the coordinate neighborhood V can be chosen
small enough so that q(O)  V consists only of the single slice q(W),
and thus U  P == -((W)). It follows in this case that P has the
relative topology; hence (P,i) is an imbedding. The uniqueness of the
manifold structure in this case is guaranteed by result (b) of 1.33.
Examples
d
The differential of the function f(p)==  r(p) 2 on II d is surjectivc
t--1
except at the origin. Thus it follows from 1.38 that the spheref-(r2),
for a constant r > 0, has a unique manifold structure for which it is a
submanifold of I  under the inclusion map. In particular, this is the
same manifold structure as the one defined in Example 1.5(d).
We define a map  from the general linear group Gl(d,,) (Example
1.5(f)) to the vector space of all real symmetric d × d matrices by
(A) = AA ,
where A t is the transpose of the matrix A. Let
O(d) =
where 1 is the d × d identity matrix. O(d) is a subgroup of Gl(d,)
under matrix multiplication called the orthogonal group. To apply
1.38 to conclude that O(d) has a unique manifold structure such that
(O(d),i) is a submanifold of Gl(d,), and that in this manifold struc-
ture i is an imbedding and O(d) has dimension (d(d- 1)), one

4 Manfold
need only check that d, is surjective at  ¢  O(d). For this, it
sulEc to check that d,p is surjective, since wlnev ¢  O(d),
vp == vpo r.
where ro (right WlIon by o) is the difl'eomorphism of Gi(d,)
defined by r,(') == . We leave the details to the reader as an exercise.
VECTOR FIELDS
1.41 Deflnltiom Smooth curves o: (a,b)--M and their tangent
vectors (t) were defined in 1.23(e). We say that a mapping o: [a,b] --,. Mis a
smooth curve in M if  extends to be a C '° mappingof (a -- e, b + e) into
M for some  >, 0. The curve o: [a,b] -- M is said to be p/ecew/. oth
if there exists a partition a -- u < o 1 <-.. < o, == b such that  ] [ ,+1]
is smooth for each i == 0,..., n- 1. Observe that piecewise smooth
curves are necessarily continuous. If o: [a,b] -- M is a smooth urve in M,
then its tangent vector
|
is well-defined for each t  [a,b].
1.42 Definitions A vector field X along a cu-'¢ ¢: [a,b]- M is a
mapping X: [a,b] - T(M) which I/fts ¢; that is, r o X ffi ¢. A vector field

Vector
X is called a smooth (C °) vector field along  if the mappin8 X: [a,b] --
T(M) is C °. A vector.Bald X on an open et U in M is a i/flng of U into
T(M), that is, a map X: U-- T(M) such that
r - X = identity map on U.
Again, for the vector field Xto be smooth (C ) means that X  C'°(U,T(M)).
The set of smooth vector fields over U forms in the obvious way a vector
space over IR and a module over the ring CO(U) of C '° functions on U.
If X is a vector field on U and m  U, then X(m) (often denoted Xm) is an
element of M,. If f is a C* function on U, then X(f) is the function on U
whose value at m is X,(f).
1.43 lroposltlon Let X be a vector.eld on M. Ten tbe followinB are
equimlent:
fa)
(b) If (U,  .... , xa) ia a coordinate system on M, and if {a}/a the
collection of functions on U defined b
(c) ,7r   ope in M J' Co(I,'), t XU'I  Co(I,')..
I,OOF (a)= (b) The fact that X is smooth implies that X I U is
smooth; and since the composition of differentiable maps is aain
differentiable, it follows that dx, o X I U is smooth. (Recall that the
dx are coordinate functions on =-x(U) c T(M), 1.25(3).) But
dx, o x l U = o,.
Hence the a, are C ® functions on U.
(b)=-(c) It suffices to prove that X(f) I UC*(U) where
(U, xx,... ,x a) is an arbitrary coordinate system on M for which
U  V. But, by (b),
x  f)l v
and the right-hand side is a C m function on U.
(c) =- (a) To prove that X is C ®, it suffices to prove that X I U is
CO where (U, Xx ..... xa) is an arbitrary coordinate system on M.
To prove that X I U is CO, we need only check that X I U composed with
the canonical coordinate functions 1.25(3) on r-x(u) are C ® functions.
Now, x, o r o X I U = x, and dx o X I U ----- X(x), all of which are CO
functions on U.

6 Mm,loI
1.44 Lie Bracket If X and ¥ are smooth vector fields on M, we define
a vector field [,Y,Y] called the L/e bracket of X and ¥ by setting
(1) [X,;%() -- X.( /) -- Y.(X/).
1.45 Proposition
(a) [X,Y]  d a m,oth vector.fld on M.
(b) If f, gC®(M), thin [f,g YJ - fg[X,Y] + f(Xg) Y- g( ¥f)X."
() [x, - -[,x].
(d) [[X,,Z] + [[',Z]'] + [[Z,X,Y] - Ofo, ,,oo
fteIds X, Y, a Z on M.
We leave the proof  an exercise. Part (d) is known as the Ya¢ob[ [dentity.
A vector space w/th a b/I/near operation satisfying (c) and (d) s called a
Lie algebra.
1.46 Definition Let X be a smooth vector field on M. A smooth curve
 in M is an/ntgm! curve of'X if
(1) o(t) - x((,))
for each t in the domain of (.
1.47 Let A" be a C ® vector field on M, and let m ¢ M. Let u now consider
the question: Does there exist an integral curve of X through m, and if so,
is there a unique one ?
A curve ,: (a,b) --* M is an integral curve of.Y if and only if
Let  ieet thi  contiuat. Supple t 0  (,b) d t'(O) -- .
- ox
where thef are C ® functions on U. Moreover, for each t such that y(t) ¢ U,
Thus, in view of (2) and (3), equation (!) becomes

Vector F/elds
Thus , is an integral curve of X on )f-x(U) if and only if
(5) rd7' It == f ° q:X(7x(t) .... ,7(0) (i == I,... , d and t ¢ "x(U)),
where 7 == x, o 7. Equation (5) is a system offirst order ordinary differential
equations for which there exist fundamental existence and uniqueness
theorems [I I]. These theorems, when translated into manifold terminology,
give the following.
1.48 Theorem Let X be a C ® vector fleid on a differentiable manifold M.
For each m  M there exists a(m) and b(m) in r t3 ( : oo), and a snooth curve
7m: (a(m),b(m)) -; M
(i)
such that
(b)
0  (a(m),b(m)) and 7=(0) =¢ m.
7= is an integral curve of X.
If p: (c,d)- M is a mooth curve satisfying conditions (a) and (b),
then (c,d)  (aCm),bCm)) and p = 7,, I (c,d).
We continue with the statement of the theorem after the following.
Definition
(2)
by setting
(3)
For each t ¢ , we define a transformation X, with domain
 .ffi {m  M: t ¢ (a(m),b(m)))
x,(m) = 7.,(t).
(d) For each m  M, there exists an open neighborhood V of m and
an e > 0 such that the map
(4) (t,p)
is defined and is C® from (--e,e) X V into M.
(e)  is open for each t.
(f) J , =ffi M.
>0
(g) X :  " _ is a diffeomorphism with inverse X_.
(h) Let s and t, be real numbers. Then the domain of X, o X is contained
in but generally not equal to ,+. However, the domain of X, o X,
is ,+ in the case in which s and t both hare the same sign. Moreover,
on the domain of X, o X we have
x, o ffi

Manfo/ds
ooF We let (a(m),b(m)) be the union of all the open intervals
which contain the origin and which are domains of integral curves of
X satisfying the initial condition that the origin maps to m. That
(a(m),b(m))  e (and hence part (f) holds) follows from an application
of the fundamental existence theorem [11, 'lto 4, p. 28] to the
system 1.47(5). Now if et and # are integral curves of X with domains
the open intervals A and B (with A  B # ), and if et and # have the
same initial conditions or(t0) -- (t0) at some point to ¢ A  B, then the
subset of A r B on which e and fl agree is nonempty, open by the basic
uniqueness theorem [11, Tllitll 3, p. 28], and closed by continuity;
and hence this subset is equal to A  B by the connectedness of A  B.
It follows that there exists a curve Ym defined on (a(m),b(m)) and
satisfying parts (a), (b), and (c).
The existence of an e > 0 and a neighborhood V of m such that the
map (4) is defined on (--e,e) × Vis the content of rt.otr 7 on p. 29 of
[11]. That the map (4) is smooth (and hence part (d) holds) follows
from THmteu 9 on p. 29 of [11] on the differentiability of the solutions
of 1.47(5) with respect to their initial values.
Next we prove part (h). Let t  (a(m),b(m)). Then m -. y,(t + m)
• is an integral curve of X with the initial condition 0  ym(t) and with
max/real domain (a(m) -- t, b(m) -- t). It follows from part (c) that
C6) (aCm) - t, bCm) - t) = (a(ymCt)),b(y, Ct))),
and for s in the interval (6),
(7) yv,,c0(s) z y,(t + s).
Now let m belong to the domain of X0 o Xt. Then t ¢ (a(m),b(m)) and
and (5) follows from (7). It is ey to construct exnmples to show that
the domain of X o X is generally not equal to ,+. (Consider, for
example, the vector field ]r: on s _ 0) with s z --1 and t -- 1.)
If, however, s nd t both have the stone sign, and if m ¢ ,+, that is,
if s + t ¢ (a(m),b(m)), then it follows that t ¢ (a(m),b(m)) and, by
(6), s ¢ (a(7,(t)),b(7,(t))) hence m is in the domain Of X0 o X.
Prts (e) and (g) re trivial if t  0, so nume that t , 0 and that
m¢t. (A similm" m'gnment will prove (e) and (g) if t  0.) It
• follows from irt (d) nd the comlctness of [0,t] that there exists
 neighborhood W of y,([0,t]) and an e  0 such that the map () is
defined and C ® on (--e,e) × W. Choose • positive integer n lm-ge
enough so that tin ¢ (--e,e). Let  -- X/, I W, and let W: -- u-:(W).
Then for  -- 2 .... , n v inductively define
and

Vector Fields 39
g is a C  map on the open set Wx  W. It follows that W, is an
open subset of W, that W, eontains m (since if X,/, composed with
itself n times is applied to m, we obtain y,(t), which lies in W), and that
by part (h),
Consequently, W. = ,; hence , is open, which proves part (e).
Finally, X, is a 1:1 map of ., onto _, with inverse X_,. That
X, is C ® (similarly for X_,) follows from (8), which locally expresses
X, as a eomposition of C ® maps. Hence X, is a diffeomorphism from
, to _,, which proves part (g) and finishes Theorem 1.48.
1.49 Definitions A smooth vector field X on M is complete if  . M
for all t (that is, the domain ofT'. is (--oo,oo) for each mM). In this
cse, the transformations X, form a group of transformations of M para-
metrized by the real numbers clled the l-parameter group of X. If X is not
complete, the transformations X, do not form a group since their domains
depend on t. In this cse, we shall refer to the collection of transformations
X, as the local l-parameter group of X.
1.50 Remarks A simple example of a non-complete vector field is
obtained by considering the vector field a/at: on the plane with the origin
removed- If a > 0, the domain of the maximal integral curve through
(o,0) is (--a,+ ). In the case in which the manifold M is compact, .any
C  vector field on M is complete. We leave the proof as an exercise.
1.51 Delinition Let p: M-- N be C ®. A smooth vector field X along
p (that is, X  C®(M,T(N)) and = o X  p) has local C ® extensions in N
if given m  M there exist a neighborhood U of m and a neighborhood
V of p(m) such that p(U) " V, and there also exists a C ® vector field '
on V such that
1.52 Remark It is easy to prove that a C ® vector field X along an
immersion p: M.-*N always has local C ® extensions in N. However, if
p is not an immersion, such extensions generally do not exist. Consider the
following example. We shall first define a smooth vector field X along a
smooth curve g: I -- I in the real line. Let
(1) (t) ffi t s,
(that is, et ffi r s where • is the cnonical coordinate function on ), and let

40 Manifolds
Since ,, is a homeomorphism, there is induced a vector field , on II so that
the above diagram commutes. Now, X is a smooth vector field along ,,,
but ,,o is not a smooth vectorfleld on [R z. To show this, let u -- t'. Then
(3)
Thus
(4)
. 3r2/, d_.
and the function r */s is not differentiable at the origin. If we extend et to be
a mapping into the plane by etting
(5) (t) = (is,0),
and again we let .Y(t) -- (t), then .Y is a smooth vector field along the
one-to-one C ® curve a in the plane, which admits no local C ® extension to
a neighborhood of (0,0).
1.53 Proposition Let m e M , and let X be a smooth vector fieid on M
such that X(m) ee O. Then there exists a coordinate system (U,) 'ith
coordinate functions xl, . . . , xe on a neighborhood of m such that
(1) x I v -
eotw Choose a coordinate system (Iz,.) centered at m with coordinate
functions y .... , y, such that
It follows from 1.48(d) that there exists an e > 0 and a neighborhood
W of the origin in *- such that the map
(t, as .... , a.) - x,(.-,(0, a, .... , ,))
is well-defined and smooth for (t, 02 .... , a,)e (--e,e)× W = *.

Ditributions and the Frobeniu Theorem
Now,  is non-singular at the origin since
:2).
Thus by Corollary (a) of 1.30, q -ffi o-* is a coordinate map on some
neighborhood U of m. Let x, .... , x, denote the coordinate functions
of the coordinate system (U, qQ. Then since
we have
X[ U .= ax
1.54 Definition Let q: M-- N be C . Smooth vector fields X on M
and Y on N are called e-related if dq o X = Y o q.
1.55 Proposition Let q: M.-N be C ®. Let X and X be smooth
vector fields onM, and let Y and Y1 be smooth vector fields on N. If X is
related to Y, and if X: is e?-related to Y: , then [X,X,] is related to [ Y, Y].
PROOF We must show that dq o [X, Xs] ffi [Y, YI] o e. For this,
let m  M andf '®(N). Then we must show that
(1) d( [X, Xz],)([) ffi [ Y, Y],o,o (.
We simply unwind the definitions:
(2) d(lX, X,l.)(f)ffi [X,X,],,,o q,)
ffi x.(Y,(f) o ) - x,l.(Y(f) o )
ffi z,(f)) -
ffi Z.,.)(nO0) - r11.,., (r  O)
= [z,
DISTRIBUTIONS AND THE FROBENIUS THEOREM
1.56 Definitions Let c be an integer, IX c  d. A c-dimensional
distribution  on a d-dimensional manifold M is a choice of a c-dimensional
subspace (m) of Mm for each m in M.  is smooth if for each m in M
there is a neighborhood U of m and there are c vector fields X ..... X of
class C ® on U which span  at each point of U. A vector field X on M is
said to belong to (or lie in) the distribution  (X  ) if X,  (m) for each

 Manfo
m  M. A smooth distribution P is called/nookaivt (or ¢on/etely int-
grab/e) if IX, Y]  P whenever X and ¥ are smooth vector fields lying in
1.$7 Definition A submnnifold (N,] of M is an /ntegral man/fo/d
of a distribution P on M if
(1) dlP(N) -- P(p(n)) for
I.8 Remarks Our object in this section is to prove that a neceary
and sufficient condition for there to exist integral mauifolds of  through
each point of h/is that P be involufive. Perhaps a word of explanation is
in order about the expreion "'completely lntegrab/e" sometimes used in
place of "involutive." We have required integral manifolds to be submnnl-
folds whose tangent space coincid¢ with the subspace determined by the
distribution. One could define a weaker notion of integral manifold by
requiring only that the tangent spaces of the submnnifold be contnined in
but not necestrily equal to the distribution at each point. It is possible for a
distribution  to be "integrable" in the sense that it has low-dimensional
"integral manifolds," but not completely integrable in
does not have integral manifolds of the maximal dimension. For us, unless
specified otherwise, integral manifolds of distributions will always be taken
to mean integral manifolds of maximal dimension, that is, as defined in 1.S7.
1.$9 laropoition Let  e a smooth distribution on M such that through
each point of M there passes an integral manifold of . Then  L involutive.
PROOF Let X and Y be smooth vector fields lying in , and let m ¢ M.
We must prove that IX, Y], ¢ (m). Let (Ar,p) be an integral manifold
of through m, and suppose that p(no) --m. Since tbp: Ar - (p(n))
is an isomorphism at each n in A r, there exist vector fields ',
such that
() do',Xo,
do 1, Yo.
It is easily checked that ' and J are smooth. By I.S, [',  and
IX, Y] are related. Hence IX, Y]., -- d([', ,,,) (m).
1.60 Theorem (Frobonlus) Let  be a c-dmenonal, nvove, C ®
distvOuton on M . Let m  M. 77n there exists an ntegral manifold of
 passn through m. Indeed, there exists a cubic coontOmte system (U,q)
which is centered at m, with coordinate functions x, . . . , xs such tat the
slices
(1) x -- constant for all 1
ave integral manifolds of ; and if (N,) ia a connected ntegval manifold
of   tt ,(N) = U, ten N) Iea n one of these slices.

Distributions and the Frobenha Theorem 43
Iqoov We shall prove the existence part of the theorem by induction
on ¢. For the case ¢ -- 1, choose a vector field X lying in , defined
on an open neighborhood of" m, such that X(m)  O. Then Proposition
1.53 yields a coordinate system (U,?) about m, which can be taken to
be cubic centered, for which X [ U -- /xs. Hence the theorem holds
for c -- 1.
Now assume that the theorem holds for c- 1; we prove it for a
distribution  of dimension c. Since  is smooth, there exist smooth
vector fields XI,..., X spanning  on a neighborhood  of m. By
1.53, there exists a coordinate system (V, ys,... ,Y) centered at m,
with V = , such that
¢2) x, I v - --.
On V, let
]ramX ,
(3) r, - x, - x,v)Xl 0 - 2, ...,
Then the vector fields YI,.-., Y, are independent C ® vector fields
spanning  in V. Let S be the slice yl -- 0, and let
(4) Z,= ¥,IS (i = 2,... ,c).
Then since (2) and (3) imply that
(5) r,(y,) = o ( = 2 .... , c),
the Z are actually vector fields on S; that is, Z(q) S, whenever
q  S. The Z, span a smooth c- l dimensional distribution on S.

44 Mtmlfold
We claim this is involutive. Indeed, the Z, are /-related (inclusion
map of $ in M) to the ¥, and therefore, by 1.55, their Lie brackets
are also i-related to the corresponding brack.,ets of the . But [ Y, Y],
(i,j  2), has no component in the Yt on (apply toy, and get 0).
Therefore, there exist C" fanetiom ca. such that
(6)
on V, and thus
(7)
This proves that the distribution on $ is involutive. By tl induction
hypothesis, there exists a centered coordinate system wl .... , w.,
on some neighborhood ofm in $ such that the slices defined by w -,
smut for all i  {c + 1 ..... d) are precisely the integral manifolds of
the distribution spanned by Zs,..., Z, on this neighborhood.
The functions
where r: V--,. $ is the natural projection in the y coordinate system,
are defined on some neighborhood of m in M, are independent at m,
and they all vanish + at m. Thus there is a cubic-centered coordinate
system (U,q) with the coordinate functions x .... ,x, on a suitable
neighborhood U of m. We now prove that
(9) Y(x,+,) m 0 on U (l -- 1,..., c; v -- 1,..., d-- c).
From this it follows that the vector li¢lds 818x .... ,8lx, form a
basis for  at each point of U, and thus the slices (1) are integral maul-
folds of .
To prove (9), tlnt observe that (8) implies that
(10) x,./1 (] - 1)
ey, 10 (]-2, .... d)
on U; and thus (2), (3), and (10) imply that
(1) Y "x our3,
so certainly (9) holds for l- 1. Now let /e{2,... ,c} and
v{l ..... d--c}. By (11),

Dixtributlons and the Frobenius Theorem 45
The involufivity of  implies that there are C ® functions ¢ such that
(13) [Yx,] -- cY.
Using (13), (12) becomes
•
(14)
Slioe X  mtat,...,X,  goner
Fix a slice of U of the form x, == constant .... , x == constant. On
such a slice, ¥,(xa+,) is a function of xx alone, and (14) becomes a
system of c -- I homogeneous linear differential equations with respect
to xx. Such a system has a unique solution with gven initial values
[11]. Since the system is homogeneous, the 0 functions give a solution.
But each such slice has a unique point in S  U, and on S  U,
(Is) ¥,(xa+,) - z,(w.+,) - o (i - 2 .....
The first equality follows from (4) and (8), and the second from the
fact that the integral man/folds of the distribution on $ determined by
the Z, are given by suitable slices in the w coordinate system. It follovs
from (14) and (15) that the functions Y(x,.,) must be identically zero
on U. Thus (9) holds, and the induction step is completed.
Finally, suppose that (',W) is a connected integral manifold of @
such that p(N) c U. Let ,r be the projection of fl' onto the last d -- ¢
coordinates. Then vectors in @ are annihilated by d(,r o q). Thus
for each n • N. By 1.24, r o q o p is a constant map since N is connected.
Thus W(N) is contained in one of the slices (1).
1.61 Remarks The classical version of the Frobenius theorem appears
quite different from our version in 1.60. The classical Frobenius theorem
can be formulated as follows.

46 Manlfold
Let U and V be opun sets in " and - respectivdy. We use ¢onrdinates
rt,. • •, r, on " and at, .... a, on -. Let
(t) b: U
be a C" map of U x V into the set of all n x m real matricas, and let
(r,a,)eu x v. If
b, b,._.
U-- I .... ,n;,,O-- I .... ,m)
on U x V, then there exist neighborhoods J, of r in U and V, of , in V
and a unique ¢® map
(3) :)x
such that if
.(r) - ,(r.s) 0  v,. r  u,).
then
(4) ,(r,) - s.
d,, I, --
for all (r..,)  , x V,.
Equation (4) b a sl total de! tl. We sify in (1)
wt e diffenflM  a p ld   a fusion of e ph, and in
(2) we ve a  d st cdition for e esn of such
a p  e sfi ifial condifi. It n  own at is veion
is uivalent to 1.. For emple, if we s wi a cimsionM, involu-
rive, C = disbufion  on M  and a int m e M, tn we n obn !.
from e classi vei  follows. We can fit ch a crna
sm (W,) aut m with cinate funions Yx .... ,y d with
• (- U g V   g , for wch the exist  funions , on
U g (i -- !,..,, c; j -- 1 .... , d-- c) such that the vor fiel
() -+ - (- .....
.  By,+#
s   . en we dene a map b  in (1) by tdng
() " b(ra) - ,(rj)}.
It ms out t e involuvity of  impli tt (2) is mtb, and fom
• e p ¢ e  n e di Nim sysmm 1.(1). Convly,
e n obn  clasd veion from I. in a similaz y.
We shall give in Chapter 2 yet another version of the Frobenius theorem
in terms of differential forms and differential ideals.

Distributions and the Frobenius Theorem 47
In 1.32 we considered the s/tuafion in which a C ® map o: N---M
factors through a submanifold (P,) of M so that o =  o o. where
oo: N -- P, and we sought sufficient conditions for o. to be C ®. An impor-
tant case occurs when (P,) is an integral manifold of an involutive distribu-
tion on M.
1.62 Theorem Suppose that p: N --*, M  Is C ®, that (P, ) is an Integral
manifold of an involutive distribution  on M, and that o factors through
(p,q), that i, p(N) c (p). Let p: N-- P be the (unique) mapping such
that  o ,o  '.
(1)
Then too is continuous (and hence C ® by 1.32(a)).
,tooF Let p belong to an open set U in P, and let n ¢ 0e-x(p). Using
1.60, we can obtain an open set 7 withp ¢ 7 c U and a cubic coordinate
system (V,) centered at q(p) with coordinate functions
such that the slices
(2) x, = constant for all i ¢ {c + 1 .... , d}
are the integral manifolds of  in V, and such that q(7) is the slice
(3) x+l = "'" = x, = 0.
-I(V) is open in N. Let W be the component of t0-x(V)containing n.
W is open. To prove that t0o is continuous, we need only prove that
t%(W) = 7 = U. By the commutativity of (1) and the injectivity of
it suffices to prove that t0(W) lies in the slice (3) of V. Now, t0 is contin-
uous and W connected; hence (W) is connected. Moreover, (W)
has at least the point t0(n) in common with the slice (3). So since
t0(W) lies in a component of q(P) r V, it is sufficient to prove that
components of q(P) r V are contained in slices of the form (2).
Let C be a component of q(P) r V, and let r: V-- *- be defined
by
(4) ,(m) = (x,+,(m) ..... x,(m)).
Then since P is second countable, and sine q(P) r V is a union of
the slices (2) due to the fact that (P,q) is an integral manifold of
it follows that r(q(e) t V) consists of a countable number of points
in [*-'. Thus r(C) is a connected countable subset of [-; hence
r(C) is a single point, and C lies in a single slice.

48 olds
1.65 l:-finition A maximal integral man/fo/d (/g,) of a distribution
 on a mnifold M is a conneCd integral manifold of who imase in M
is not a proper subt of any other connected integral manifold of . That
is, thero does not et a conneCd integral fold (Nt,lp) of  such tlt
(N) is a proper subt of (N).
1.64 Theorem Let  be a c-dimensional, involutive, C ® distribution on
M . Let m  M. Then through m there passes a unique maximal connected
integral manifold of , and every connected integral manifold of  through m
is contained in the maximal one.
rooF Existence Let K be the set of all those points p in M for which
there is a piecewise smooth curve joining m to , whose smooth portions
Intoal ¢ugv of D
are l-dimensional integral curves of , that is, the/r tangent vectors
belong to . By 1.60 and the cond countability of M, there/s a
countable coveting of M by cubiccoordinate systems {(U,, x1', • • •,
i -- 0, I, 2 .... } such that the integral man/folds of in U, are the slices
(1) x+ -- constant for allj ¢ {1 .... , d -- c}.
We shall assume that m ¢ U s.
How let p ¢ K. Then there exists an index i, such that p ¢ U,,,
and there is a slice $, of U, of the form (1) containingp. Observe that
$, c K. It follows from 1.60 that the collection of all open subsets
of all such $, as p runs over K forms a basis for a locally Euclidean
topology on K, and that we obtain a differentiable structure on K if we
take the maximal family of coordinate systems (with respect to 1.4(b))
containing the collection
We claim that K with this topology and differentiable structure is a
connected differentiable manifold of dimension c. Kis clearly connected
since it is pathwise connected by construction. We have only to prove
that the topology on K is cond countable. For this, fix an
i ¢ (0, 1, 2 .... ). We need only show that there are at most ¢ountably
many slices of U, in K. Each point of U which lies in K is joinable to
m by a piecewise smooth curve whose range also lies in K. To each
such curve from m to points in U there corresponds (although not
uniquely) a finite sequence
(3) rY,, rY,,,... , rY,.,

Distribution and the Frobenius Theorem 49
of the coordinate neishborhoods throu8 h which the curve passes in
order. The curve thus besins in the slice of U. containing m, passes
throush some slice of U, !, then throush some slice of U,,, and so on,
until in a finite number of steps it reaches a slice in U,. Since there are
at most countably many such sequences (3) from U0 to U,, we need
only show that for each such sequence there are at most countably
many slices of U, reachable in the above manner. For this, we need
only observe that for any J, k  (0, 1,...)
a sinsle dice of U can intersect at most
countably many slices of U; for if S ..__J
is a slice of U, then S  Ut is an open
submanifold of S and therefore consists
of at most countably many components,
each such component being a connected '
integral manifold of  in Ut, and hence
lying in a slice of Ut. This proves the
second countability of K.
(K,i), where i: K-- M is the inclusion map, is now a submanifold
of M and is a connected integral manifold of  passing through m.
Moreover, (K,i) is a maximal connected integral manifold of . Forlet
(N,p) be any connected integral manifold of
and let pp(N). There is a piecewise smooth curve ¢: [0,1]--V
joining p-l(m) to W'l(p). (Connected manifolds are pathwise connected.)
Then p o c is a piecewise smooth l-dimensional integral curve of
connecting m top. Thusp e K, and so (N) = K, which proves that K
is maximal. Thus we have proved the existence of a maximal con-
nected integral manifold (K,i) of  passing through m, and have proved
that every connected integral manifold of  through m has its image
in K.
Uniqueness (el. 1.33) Let (N,,p) be any other maximal connected
integral manifold of passing through m. As we have observed above,
p(N) = K; thus p factors as follows:
N  )M
Po is C ® by 1.62, and is 1 : 1 and non-singular since p is 1 : 1 and non-
singular. Pe is onto since the fact that (N,p) is a maximal integral
manifold of  through m implies that p(N) cannot be a proper subset
of K. It follows from Corollary (a) of 1.30 that Pe is a diffeomorphism.
Thus (N,p) and (K,i) are equivalent, and the maximal connected integral
manifold of  through m is unique.

1 Prove that in Example 1.5(d) one does indeed obtain a differentiable
2 The usual different/able structure on the real line I was obtained
by taking £r to be the maximal coIletion containing the identity
map. ,Wz be the maximal collection (w/th respect to 1.4(b))
containing the mp t--,t . Prove that £r # £ri, but that (')
and (,-'t) are difeomorph/c.
3 Let {U.} be an open cover of a man/fold M. Prove that there ex/sts
a refinement { F.} such that . c U. for each
4 Use the fact that mnifolds are regular and paracompact to prove
that manifolds are normal topolosiud spa.
s prove 1.25(a), (b), and
6 Prove that if p: M--V is C ®, one-to-one, onto, and everywhere
non-s/ngular, then p is a difeomorphism. (This propm/tion depends
strongly on the tw.ond countability of M. Here is u outl/ne for a
proof. Tiffs map p is a d/feomorphism if and only/fd is surjective
everywhere. If ap is not surjective at some point, then the d/memion
of Mis less than the dimension Of N'. Let d/m M --panddim N"
Assuming that p < , one arrives at a contrad/ction as follows.
Let (U,q) be a coordinate system on V such that q(U) --
Sine ,p maps M onto N, the range of ? o p is all of It'. Now we
show that this yields a contradiction. One way is to use 1.35 to
observe that the range of q o  is a countable union of nowhere
dense sm in ', and therefore, by the Baire category theorem,
could not possibly be all of IR'. Another way is to use the fact that
p < d to prove that the range of ? o p has measure 0 in ', which
also is a contradictioa (where a set in ' has measure 0 if it can be
covered by a sequence of balls, the union of whose volumes is
arbitrarily small). That the range of q o  has measure 0 in
follows from the second countability of M and the fact that a C
mp " -- ' hu image of measure 0, and this, in turn, follows
from the fact that a Ca map IR * - It • takes sets of measure 0 to
sets of measure 0. To prove the latter, observe that if f: 
is Ca, and if A is a compact set in IR', thenfhas a Lipschitz constant
/Con A,
so thatfmagnifles the volume of balls in A by at most/(', and hence
takes sets of measure 0 in A into sets of measure 0.)
7 Prove 1.33(a) and (b).

Exerc/se $I
g Obtain the classical implicit function theorem 1.37 from 1.38.
9 Letf: R,  R be defined by.
f(,,y) - x, + ,y + y* + i.
For which points p -- (0,0), p = (t,t), P = (--t,--t) is f-ffi(f(p))
an imbedded submanifold in Rt ?
10 Let Mbe a compact manifold of dimension n, and letf: M--*. R be
C ®. Prove thatfcannot everywhere be non-ingular.
II Find counterexamples to show that Proposition 1.36 would fail
if either of the hypotheses closed or imbedded were deleted. In
fact, one can prove more; namely, if (M,o) is a submanifold of
N such that whenever g¢ C®(M) there is a C  function f
on N such thatfo 0 = g, then 0 is an imbedding and 0(M) is closed
inN.
12 Supply the details for 1.40(a) and (b).
13 Prove Proposition 1.45.
14 Is every vector field on the real line complete?
15 Prove that if (U, xffi .... , x,) is a coordinate system on M, then
16 Let N = M be a submanifold. Let 7: (a,b)-,. M be a C ® curve
such that 7(a,b) = N. Show that it is not necessarily true that
(t)  V<,> for each t  (a,b).
17 Prove that any C '° vector field on a compact manifold is complete.
18 Prove that a C ® map f: ' - x cannot be one-to-one.
19 Supply the details of the equivalence of the two versions 1.60 and
1.61 of the Frobenius theorem.
20 Let q: N- Mbe C ®, and let Xbe a C ® vector field on N. Suppose
that d(X()) ffi d(X(q)) whenever (V) m (q)- Is there a smooth
vector field Y on M which is related to X?
21 The torus is the manifold 5  x 5 . Consider 5  as the unit circle
in the complex plane. We define a mapping q: [ --* 5  x 5  by
setting q(t) = (e'",e' where g is an irrational number. Prove
that ([,?) is a dense submanifold of 5  x S x. This submanifold
is known as the skew line on the torus.
22 Let 7(t) be an integral curve of a vector field X on M. Suppose
that (t) = 0 for some t. Prove that 7 is a constant map, that is,
its range consists of one point.

A Riemannian structure on a different/able man/fold M is a smooth
choice of a positive definite inner product ( , ), on each tangent
space M,, smooth in the sense that whenever X and Y are C
vector fields on M, then (X, ]Q is a C ® function on M. Prove that
there exists a Riemannian structure on every d/fl'erentiable man/Fold.
You will need to use a partition of unity argument. A R/enumn/an
manifold is a differentiable manifold together with a Riemannian
structure.
Consider the product manifold M < Nwith the canonical projections
rx: M × N- M and vrs: M x
(a) Prove that :-.-M x N is C  if and only if vrzo and
(b) Prove that the map v -. (dvrz(v),dvr,(v)) is an isomorphism of
(M × N),. with M,
(c) Let X and ¥ be C  vector fields on  and V respectively.
Then, by (b), X and ¥ canonically determine vector fields
l' = (x,o) and f' - (0, Y) on r x 7V. Prove that [l', F] = 0.
(d) Let (n,n.)¢ M x N, and define injections
and 1.: V-.- M x N by setting
let v ¢ (M × N)....0, and let
dr,(v) ¢ N,. Let f ¢ C®(M x N). Prove that

There are a number of vector spaces and algebras naturally associated with
the tangent space Mm. Suitably smooth assignments of elements of these
spaces to the points in M yield tensor fields and differential forms of various
types. We shall first develop some of the pertinent facts from multilinear
algebra, and then beginning with 2.14 we shall apply these concepts to
manlfolds.
TENSOR AIWD IXTER/OR ALGEBRAS
Throughout 2.1-2.13, F, IF, and U will denote finite d/mensional real
vector spaces. As usual, F* will denote the dual space of F cons/sting of all
real-valued linear functions on F.
2.1 Detin/tions Let F(I,',FtO be the free vector space over IR whose
generators are the points of I/× IF. Thus F(',W) consists of all finite
linear combinations of pa/rs (v,w) with v  I/and w  IF. Let R(I,FtO be
the subspace of F(I/,W) generated by the set of all elements of F(I/,W)
of the following forms:
O)
(v + v,, "9 - (v,.,) - (v,,)
(v, .'x + .',) - (v,.,O - (v,,)
(av,.,) - a(v,)
(v,a.,) - a(v,.,)
a
W, WZ, W| 
The quotient space F(',W)IR(V,W) is called the tensor product of V and
W and is denoted by V ® W. The coset of g ® W containing the element
(v,w) of F(I/,W) is denoted by v ® ,,. It follows from (1) that we have the
following identities in g ® W:
(vz + vs)®., -- vz ®., + v,® ,,
v® (.,z +
a(v ® .,) -- av ®.,  v ® a.,.

Ten, or and Exterior Algebra 55
2.2 The following properties of the tensor product are easily established,
and are left to the reader as exercises.
(a) Universal Mapping Property. Let p denote the bilinear map (v,w)-
v ® ,, of V × W into V® W. Then whenever U is a vector space
and l: V × W-- U is a bilinear map, there exists a unique finear
ma/ ': V ® W-- U such that the following diagram commutes:
V®W
(1) " ""-
V×W . U
The pair consisting of V ® W and p is said to solve the universal
mapping problem for bilinear maps with domain V × W. Moreover,
V ® W and p are unique with this property in the sense that if X
is a vector space and ?: V × W--*X a bilinear map with the above
universal mapping property, then there exists an isomorphism
z: V ® W-, X such that  o p  .
(b) V ® W is canonically isomorphic with W ® V.
(c) V ® (W ® U) is canonically isomorphic with (V ® W) ® U.
(d) By property (a), the bilinear map of V* × W into the vector space
Horn (V,W) of linear transformations from V to W defined by
(f,w)(v) f(v). w forf¢ V*, v  V, and ,, ¢ W determines uniquely a
linear map z: V*® W-,Hom (V,W). z is an isomorphism.
As a consequence,
(2) dim V ® W = (dim V)(dim W).
(e) Let {e: i= I,... ,c} and {f:j = 1 .... ,d} be bases for V and W
respectively. Then {e®f:i=l ..... c and j= 1 ..... d} is a
basis of V ® W.
The tensor space Vr., of type (r,s) associated with V
2.3 Definitions
is the vector space
V®'"®V ® V*®'"® V*.
r ople. • ople.
The direct sum
(2) T(V) =  v,., (r, s > o),
where Vo.o  IR, is called the tensor algebra of V. Elements of T(V) are
finite linear combinations over IR of elements of the various Vr.. and are
called tensors. T(V) is a non-commutative, associative, graded algebra
under ® multiplication, where if u
belongs to V,., and v ffi v @'" @ W, @ v @" • @ v,*, belongs to V,,.,,,

$6 Tensors and Differential Forms
then their product u ® v is dzfined by
and belongs to Vrt+rs.,,+, .. Tensors in a particular tcrmor space Vr., are
called homogeneous of degree (r,$). A homogeneous tensor (of degree
(r,s), say) is called decomposable if it can be written in the form
vx®"" ® v®v'®-"®v,*
where v  V (i  1, .... r) and v'  V* (j  I .... , ).
oo
2.4 Definitions We let C(V) denote the subalgebra  V.o of T(V).
Let I(V) be the two-sided ideal in C(V) generated by the set of elements of
the form v ® v for v  ', and set
O)
It follows that
T,() = () c v,.o.
(2) (v) - ,(v),
and is a graded ideal in C(V). The exterior algebra A(V) of V is the graded
algebra C(V)II(V). Ir we set
(3) A(v)= v.d1(v) (/c > 2), AoCV) = 4, A,(v) = v,
(4) A(V) = A(V).
We shall dnot muRipHcation in the algebra A(V) by ^. This is called the
wedge or exterior proat. In particular, the residue class containing
vx@"" ® v is v ^.-. ^ v).
2.$ Dflnltlon A multilincar map
(I) h: V" x .-. x V'--,. W
is called alternating 
(2) ' h(v,() ..... v,(,)) ,= (sgn r)(v ..... v,) (v ..... v, e V),
for all permutations r in the permutation group $, on r l)ttors. Sgn r is
the sign oftbe permutation r (%1 ifr is even, --I ifr is odd). The vector
space of all altornating multilinemr functions
F'x .-. x
will be dnoted by A,(V), and for conv)ni)nce w) set Ao(V) = .

Tensor and Exterior Algebra $7
The following properties of the terior algebra ' left to the reader
r u ¢ A(V) and v ¢ A(V), then u ^ v  A(V) and u ^ v ,= ( -- l)v ^ u.
If el .... , e is a basis of P', then
O) {e.}
is a basis of A(V), where @ runs over all subsets of {I,..., d}, including
the empty set; where e. =eq ^"" ^ e, with lz ("" < 4 when
q> is the subset (ix,... ,4} of {1,... ,d}; and where eo =, 1 when
• ffi . In particular,
A+,(V) ffi {0)
Moreover, it foilow that
(Hint:
(j> 0).
dim A(V) -- 2 ,
( d (O < k < d).
dim At(V) -- " k!(d --
Observe that the elements {e.) span A(V). To prove that they
also are linearly independent, first prove that ea ^" "- ^ e is not zero
in A(V). For this, one must show that ea @'-- @ e does not belong
to I(V). Express an arbitrary element of I(V) in terms of the basis
vectors ex .... , e, and show that it could not equal ex @ "" @ e.
Then for finear independence of the entire set (e.), multiply the equa-
tion  ae. -- 0 by suitable products of the e to land in A(V), and
conclude that the various a. are all zero.)
Universal Mapping Property. Let ¢p denote the mapping (t,..., vt) -
v^---^v of V×'" × V(k copies) into A(V). Then p is an
alternating multilinear map. Now to each alternating multilinear
map h of V × "" × V (k copies) into a vector space _W, there corre-
sponds uniquely a linear map : At(V)- W such that o p ffi h.
A(V)
V x --. × V "'" W
The pair consisting of At(V) and p is said to solve the universal aapping
problem for alternating multilinear maps with domain V × • "" × V;
and this is the unique solution in the sense that if X is a vector space
and : V × ... × V- X an alternating multilinear map also possess-
ing the universal mapping property for alternating multilinear maps

with domain V × .-. × It, then there is an isomorphism t: A(I 0 - X
such that t o  1 .
In the special case in which W 1 , the diagram (4) establishes a
natural isomorphism
of A(V)* with the vector space A(IO of all alternating multilinear
functions on V × ... × It (k copies). It follows from property (b)
that A(IO -- {0} for k > dim It.
We shall now consider various dualities between the spaces V,..,
A( IO and the corresponding spaces (V*),.., A(V*), A(V*) built on the dual
spaee V* of V.
2.7 l:flnltion Lt It and W b real finite dimensional vtor s.
Airingof rand Wisa bilinmp ( ,): Vx WR. A g
is 1 nosinlar if wh,n w # 0 in W,    ,lent v  V
su t (v,w)#O, d wh v#0  V,  es  t
w  W ch at (v,w) # O.
 V d W  noninly  by ( , ), d fi
O) : V   by (v)Cw) -- (v,w) (v  V; w  .
It follows at 9 is 1 : 1. Sly,  B a 1:1 mp  V*. ,o
wi W*. m a non-s# $ of V d W in a o y
#ds  imoMsm 9:  W*  slly  imo W *.
2.$ lltton A non-singular parng of (V*),.o with It,.,. This pairing
is to b the biliner map (V*),.0 x V,.0  R which on decomposable d-
mllt$
v* - v' ®"" ® v,* ® u,, ®-.- ® u,+.  (V*),..
and
u - ux ®"" ® u, ® v,*+ ®'" ® v,%.0 
yields, the following:
0 ) . ( v*,u ) - v( u,) . . . v *,+,( u,+,).
It is eamily checked that there is a unique suqh bilirar mp and that it is a
non-singular pairing. This pairing establishes an isomorphism
(2) (v*),., , (v,.,)*.
On the other hand, the obvious extension of the universal mapping property
2.2(a) shows that there is a natural isomorphism
(3) (v,..)* , M,..CV)

Tensor d Exterior Algebrt 9
where /,., (V) is the vector space of all multilinear functions
Vx ." x Vx V" x ... x V"-,. IR.
Under the isomorphism (3), if ]  (V,..)*, then the corresponding multi-
finear function h in Mv.,(V) satisfies
(4) h(v,..., v,, v',..., v,') - ?;(v ®-.. ® v, ®  ®... ® v,').
Finally, from (2) and (3) we obtain an isomorphism
( (v*)r., _ M,.,(v.
2.9 Definition A non.singular pairing of At(V*) with At(V). This
pairing is to be the bifinear map of At(V*) × A(V)--* I which on decom-
posable elements v* -- v' ^-'- ^ v* in At(V*) and u -- u ^'-" ^ ut in
At (V) yields
(1) (v;u) - det(v,'(u)).
Again it is easily checked that there is a unique such bilinear map and that
it is a non-singular pairing. For k -- 0, the pairing is simply multiplication
of real numbers. This pairing estabfishes an. isomorphism
(2) At(V*)  At(V)*.
Composing (2) with the natural isomorphism
of 2.6(5) yields an isomorphism
(4)
Using the isomorphism (2), and observing that the dual space of a finite
direct sum is canonically isomorphic to the direct sum of the dual spaces,
we obtain isomorphisms
(5)
and from (3) we obtain an isomorphism
(6) A(V)*  l(v) = l(v).
Henceforth we shall make use of the identification (5) of A(V*) with A(V)*
via the pairing (1) without further comment.
2.10 Remarks on 2.9
(a) If {e ..... e} is a basis of V with dual basis {7 .... , ) in V*, then
the bases {e®) and {7®) defined in 2.6(b) are dual bases of A(V) and
A(V*) under the isomorphism 2.9(5).

(b) In 2.9(5) and (6), the following isomorphisms were established:
The second of these is the natural isomorphism arising from the universal
mapping property 2.6(c). The first, call it a for the moment, arose
from our choice of the pairing 2.9(1). There is another pairing in
common use, this one obtained by replacing 2.9(I) by
(1)
_.1 det (v(u,)).
(v*,u) - k!
This pairing gives a different isomorphism of A(V*) with A(V)*,
call it . Now A(V*) is an algebra under wedge multiplication. Via the
above isomorphisms we obtain two algebra structures A e and A t on A(V).
It is not difficult to see that iffAp(V)and g ,4,(V), then these
induced algebra structures on A(V) take the form
(2) f A, g(v ..... v)
,¢ shuffles "
and
(3) f A t g(v ..... V,+¢)
( P + q)
He a ution  • S is cI a "p, q e" if (1)
(2) <... < (p) and  + I) <'" <  + q). It follows from
(2) and (3) tt
(4) f . g - ( + q) f 
pq g"
For mple, consider the  in wchp  q  1.  y and  long
to F*  Jt(, and let v, w  F. en y A,  and y At   At(F) 
whereas
y A.  (v,,) - y(v) d(,) - y() (v),
(6) y A t  (v,,) ffi i(y(v) (,) - y(,) (v)).
We shall use only the pairing 2.9(I). It has the advantage of avoiding
factors such as the ½ in (6).

Tensor and Exterior Algebras 61
2.11 Linear Transformations of ^(V) End(^(V)) will denote the
vector space of all endomorphisms of A(V) (i.e. linear transformations
from A(V) into A(V)). Let u  A(V). Left multiplication by u is the endo-
morphism u) of A(V) defined by
(1) ,(u)v - u ^ v (v  A(V)).
The transpose i(u) of u) is an endomorphism of A(V)*. Under our identifica-
tion of A(V)* with A(V*), the transpose i(u) can be considered as an endo-
morphism of A(V*), and this endomorphism is called nterior multiplication
by u. In terms of our pairing of A(V) with A(V*), l(u) is defined by
¢.,) (,o.o,,..,+.)=(,,..,oo,+.)=o,..,,^,+.) (,,.¢v*;cv)).
If u ¢ V, then i(u) maps At(V*) into At_,(V*) for each k. In particular, if
v* ¢ V*, then itu)v*  [ and
= 0(u)v*,') = ')-
An endomorphism ! of A(V) (or in general of any graded algebra) is
(a) a derivation if l(u ^ v)  l(u) ^ v + u ^ l(v) (u, v  A(V)),
(b) an anti-derivation if l(u ^ v)  l(u) ^ v + (-- 1) P u ^ l(v)
(c) of degree k if l: As(V) -- As+t(V) for allj.
(We assume that A(V)  {0} if  <. 0.)
It is easy to see that ! ¢ End A(V) is an anti-derivation if and only if on
decomposable elements we have
(3) l(vl ^'" ^ v)  Y_ (- l)'+lv ^'-- ^ l(v,) ^...
Proposition If u  V, then i(u) i$ an anti-derivation of degree --1.
],zoo]: That the degree of i(u) is -- l if u  V is clear from the definition
2.11(2). To prove that i(u) is an anti-derivation, we check that 2.11(3)
holds. For this, it is sufficient to see that both sides of 2.11(3) give the
same result when paired with an element of A(V) of the form
, A • • • A w. That is, we must show that
O)
O(u)(v ^"" ^ vT), , ^"- ^ )

62. Tenao and Dl.O'erentlal ¥onna
Now, the left-hand  of (I) equals
(v* A--. A v?,u ^ w, A''" A w,)--
where we have put w = u. The right= ,i,d.¢ of (I) is
y (-1)'v?(u× ^... ^ ? ^.-- ', w, ^.-. ^
(Here the circumflex over a term means that it is to be omitted.)
2.13 Effect of Linear Trans/'onnations Let i: V.- IF be a. linear
transformation. Then I extends to an algebra homomorphism
0) l:
where l(vz ^-.. ^ u) =. l() ^'-- ^ l(va) and /(I) -- I.
The transpose I: IV* --* V* also extends to an algebra homomorphism
(2) t: A(W*)--- A(V*).
Our isomorphism 2.9(5) of A(V*) with A(V)* and of A(W*) with A(W')*
are natural in the sense that (2) is the transpose of (1); that is.
O) (l(.),) - (.,l@)) (,.  A(w*);   A()).
TENSOR F/ELDS AND DIFIRENTIAL FORMS
2.14 Detition
(I) ,.(f)-- U (M.),..
(2) A(M)- U A(M*..)
O) A*(M)- U A(M*.)
Let M be a differentiable man/fold. We define
tenaor bundle of type (r,s) oer M
exterior k bundle ooer M
exterior algebra bundle over M.
In the cases in wh/ch k  0 and (r,$) -- (0,0), the unions in (I) and (2) are
disjoint unions of copies of the real line--one copy for each point in M.
T,.,(M), A(M), and A*(M) have natural man/fold structures such that the
canonical projection maps to M are C ®. If (/,q) is a coordinate system
on M with coordinate functions yz ..... y,, then the bases (/Sy,} of
and {dye} of M*, for m  U, yield bases of (M=),.,, A(M*), and A(M,,*).
For. example, the basis of A(M*) is {dy, ^." ^ dye,: iz < ... < 4}.

Tensor Field and Differential Form 63
Using these bases, one can define maps of the inverse images of U in Tr.,(M),
A(M), and A*(M) under the respective projection maps to q(U)x
Euclidean spaces of the proper dimensions. By requiring these maps to be
coordinate systems, one obtains the natural manifold structures on Tr.,(M),
A(M), and A*(M) just as we did previously in 1.25 for T(M) and T*(M)
(which incidentally is simply A(M)).
2.15 Definition A C ® mapping of M into T,.,(M),A(M), or
A*(M) whose composition with the canonical projection is the identity
map is called a (smooth) tensor fieid of type (r,$) on M, a (differential) k-form
on M, or a (differential)form on M respectively. Since our tensor fields
and differential forms will always be smooth in this sense, we shall drop
the adjectives "smooth" and "differential" unless needed for emphasis.
2.16 Remarks A lifting : M--, T.,(M) is a smooth tensor field of
type (r,s) if and only if for each coordinate system (U,)h, • • • ,)') on M,
® dy ® "'" ® dye.,
(I)
where the at....#,:#t....d,e C®(U). (In the classical tensor notation one
uses lower indices on tangent vectors, upper indices on functions and
differentials, and the reverse on coefficients; thus the individual terms of
(1) would appear as
,, ,   @ dy  @ " " " @ dy ..)
a'.:'.Z
A lifting 8: M --* A(M) is a differential k-form if and only if for each
coordinate system (U,)'i .... ,),) on M,
(2)
where the b, ..... , are C ® functions on U.
2.17 Definitions -E*(M) shall denote the set of all smooth k-forms on
M, and E*(M) the set of all differential forms. EO(M) can be identified
with C®(M); indeed, the differentiable manifold Ao*(M) is simply M x
and smooth liftings of M into M × IR are simply graphs of C ® functions
on M. Forms can be added, multiplied by scalars, and given a product
(A). If ¢o, ¢p e E*(M) and c e IR, then ¢o + ¢p, c¢o, and ¢o A ¢p are the forms
which at m have the values gore + ¢pm, c¢om, and ¢o, A ¢p, respectively.
In the case in whichfis a 0-form and oJ eE*(M), we writefA ¢o simply
as f¢o. E*(M) thus has the structure both of a module over the ring C®(M)
and of a graded algebra over IR with wedge multiplication.
I II

64 Tensor$ and Differential Forms
22.18 Let ¢o c Et(M). Then o.. C A(M, and can be considered (via
the duality 2.9(4)) as an alternating multilinear function on M,.. So if
XI ..... X are vector fields on M, o(Xl, X) makes sense--it is the
function whose value at m is  ' " '
(1) (x, ..... x)(m) = o,,(x,(m), . . . , x(m)).
Thus, if we let (M) denote the C®(M) module of smooth vector fields
on M, then
(2) o: (M) × --. × (M)--* C®(M)
and is an alternating multilinear map of the
We stress that o is multilinear over the C®(M)
(3)
module 3(M) into C®(M).
module t(M); that is
o(X, ..... X_,.fX + gY, X,+, .... , X,)
- fo(X, .... X_, X, X.,...,
+ go(Xl .... , X_I, ¥, X.+, .... X)
whenever f, g ¢ C®(M) and Xx ..... X,.., X, Y, X,+x ..... Xt ¢ (M).
Conversely, it is useful to observe that am alternating C®(M) multil[near
map (2) of the module (M) into C®(M) dffines a form; for we claim that if
¢0 is such a map, then o(X ..... Xt)(m) depends only on the values of the
vector fields X¢ at m. Assuming this for the moment, it follows that
defines an alternating multilinear function o, on M,, and hence, defines
an element of At(M*); namely, given (vz ....
choose VI ..... V t ¢3(M) such that V(m)- v (- 1 .... , k), and
define
(4) ,o=(v ..... v) - o,( V ..... v,)(m).
By our claim, o.(t,, ..... vt) is well-defined, independent of the choice
of the extensions V,. Thus ¢o gives a lifting m  o. of M into A*(M),
and this is easily seen to be smooth; hence o is a form.
For simplicity of notation, we illustrate the claim with the case in which
o is a linear map of the module 3(M) into C=(M). Let X¢ (M). We wish
to show that o(X)(m) depends only on X(m). It suffices to show that
o(X)(m) == 0 if X(m) -- O. Let (U, xl ..... x) be a coordinate system
about m. Then on U, we have X- . a,(8/Sx,) where the a,(m)- O.
Now, let q be a C ® function which takes the value I on a neighborhood
V  U of m and is zero on a neighborhood of M -- U (see 1.10). Then
the vector field X, which is (8/Sx,) on U and 0 elsewhere is a C ® vector
field on M; the function  which is qa, on U and 0 elsewhere belongs to
C®(M); and
(5) x =  ', + (l - ,)x.

Tensor Fields and Differential Forms 65
Thus
So ((m)  0 if X(m)  0, m wm to  own.
Finely, we retook at via e duity 2.8(5), nsor fields n  ven
a simil indirection. If is a tensor field of ty (e,s), en we n consider
T  a map
() : (M) x ... x '(M x  x "" x ()  C(M),
which is C(M) multilinmr with st to e C(M) mul (M) and
(.
Obsee e simple fo wch formula (2) of 2.10(b)  when , 9 
( and X, Y  (M). In is c, we have
(s)   (x,  - (( - ((.
2.19 Definition lfffC®(M), the differential dfis a smooth mapping
of T(M) into IR which is linear on each tangent space. Thus df can be
considered as a l-form-, df: M-, Ax*(M). The l-form df is called the
exterior derivative of the 0-form f, and this exterior differentiation operator d
has an important extension to E*(M) given by the following.
2.20 Theorem (Exterior Differentiation) There exists a unique anti-
derivation d: E*(M)--* E*(M) of degree + 1 such that
(1) d z == O.
(2) Whenever f f C®(M) -- EO(M), df is the differential off.
PgOOF Existence. Let p  M. Let E*(p) be the set of all smooth forms
defined on open subsets of M containingp, with Et(p) the corresponding
set of/c-forms. We fix a coordinate system (U, xl ..... xd) about p.
If o,  E*(p), then
(3) o,[<dolR
where the a® f C®((domain o,) t U), where • runs over all subsets
of {1 ..... d}, and where the dx® are either dx, t ^ "" ^ dx,rwhen
{h < " " • < i,} or the constant function I when • =, . We define do,
at p by setting
C4)

We will have to show that the definition of do is independent of
the choice of coordinates. But first, we give the following properties:
() o,  (p) . do,. 
(b) do,, dnde oy on the 
where the domain of ato -F aot is (domain o  domain o),
(d) d(o ^ o,)], -- doll, ^ , + (-)., ^ do,,],
In view of properties (b) and (c), it suffgcs to check (d) for
o --fdx, ^..- ^dx,, and ¢, -- gdx, ^..- ^dx0 on some
neishborhood of p. For the case r- s- O, property (d) is
simply d(f" g)[, -- df," g(p) -F f(p)'dg, ; and the case in
which only one of r or $ is 0 is similar. Now suppose that
r  0 and $  O. If {it,..., iv}  j,... ,j}  , both sides
are O. So assume that this intersection is empty. Then
ffd,. ^... ^ dx,,) ^ (g dx,, ^... ^dx,)
-- f. g dx ^--- ^
where Iz  ---  I,÷. and • is the sign of the permutation that
has been carried out. o we have that
- ,(df, .g) ÷f(p) ,) ^ dx,.I , ^... ^
-- (df, ^ dxJ, ^..- ^ dxJ,) ^ (g(p) dx,. ^.-. ^ dx,,)
÷ (--t)'(f(p) dx,.I, ^... ^ dl,)
^ (rig, ^ dx,.I, ^'" ^ dx,.I,)
Iffis a C ® function on a neighborhood ofp, then d(df)l, -- O.
For on (domain f)  U, df -- . (flBx,) dx,, so that
Sut (a,f/ax, a,)(p) - (a,f/ax, ax,)(p), wber=m dx,I , ^ dx,[, -
--dx, I , ^ dx,I ,. So d(df, -- O.
Now we claim that the definition of d at p is independent of the coord-
inates chosen. For let d' be defined on E*(p) relative to another coordinate
system, and let o e E*(p). Then o on (domain o) ct U is given by
(3), and since d' must also mtisfy properties (a)-(e), it follows that

Ten,or k'gelds and D.'rntial Fornu 67
(s) ,r(,)l, - ,r(2 o.,,. ^-'" ^ ,,.)l, (by (b))
--
- 2 '()l,  1, ""  ,,I,
(by(d))
Now, if
of M into A*(
nborh of p, d
by pro (e) d (d). It foHo at d   tivafion of
*( of  + ng () d (2).
Uss t d'    tifion of E*(M) of de
+I fisfying (I) d (2). We tint show at
vsh on
function  wch is 0 on a neiborhood ofp d 1 on a neiborho
of M-- W. enw--wd
Now d' is d only on elemen of E*(, at is, on obly
n fos on M. We sh to dne d' on E*) for h p  M.
If w  E*), we
at p as d w. Simply I  be a C  function wch is 1 on a neibor-
hood of p d  suppo  e do of w.  w  E*(
(w is d to  0 ouide of e do of w), d w s with
w on a nborho ofp. us we  ne
and by e above rks s dnifion is inddent of e exnsion
chosen. us d'(w)l, is dned for I w in E*) and clely tisfies
propegies (a(e). (In (d), extend % A % to a fo on M by using
x A
d'(dl,  d'(d()l, 0, s d) 0 and since d' satisfi
(I) and (2).) e uations (5) now imply that whener w  E*(p),
in picul whenever w E*(, d'(w)l,  d(w)l,. s proves
Obse at it is clear from e above prf at dwl U  d(w  U)
when U is  on t in M.

2,21 Intm'lot Multlpllcatlon by Vector Fluids Let X be n smooth
vector elcl on M, and let a, e E*(M). Interior multiplication of a, by X
is the form I(X whose valu st m is the interior multiple of
(1) 0(x)l. - i(x.)(.).
That i(X)a is smooth follows easily from 2.16(2); and from 2.12 it follows
that i(X): ,(-. ,( is an anti.avation of d --1.
. Etlect of 158_appings Let : M-- N be n smooth map, and let
m M. Then we have the differential d,l,: M,,--,.N,.,, it transpose
: N,(. -- M.*, and the induced algebra homomorphism :
A(M). If  is a form on N, then we can pull o back to a form on M by
setting
(1) ()1- -
This is one of the particularly nice features of differential forms. Under a
smooth mapping, they can be pulled back from the range to the domain of
the map. Vector fields, on the other hand, do not display such pleasing
behavior-under mappings.
2,25 Propottlon Let ,: M--, N be a th p. 
(a) : E*( E*(    alebra homorph.
)  cos w d: t a, d(()) -- ) (  ,().
(c) d()(Xz, .... X(m) -- ,{,,(d(Xz(m)), . . . , X,(m))) for
  E*( for vector Xs,..., X on M.
 Rmult (c)  cl from 2.13(3); d 2.13(2) and e defi
2.(I) ply t  is  sebm homomohism. fo we chk
t ) is y a smoo fo for   E*(, obese e follow-
ed l.(d) imp at
0) d -- do V) = d().
N let  E*(, d l m  M. Ch a coorna stem
(U, xx .... , xD about N(m) d a neishbh V of m sh at
.  U. m   C  fctions  on U  at
It foH t
O) () I v-   . v dx, . V)  " "  d(x,, .
wch is a smoo gore on V. Hen ()  *(M),  us (a) is
v. To eompl ), we u

DERIVATIVE
2.24 Definition Tensor fields and differential forms can be differentiated
with respect to a vector field. The resulting derivative is known as the Lie
derivative and is defined as follows. Fix a smooth vector field X on a manifold
M. Recall (see 1.48) that we use Xt to denote the local l-parameter group of
transformations associated with X. Let Y be another smooth vector field
on M. We shall define the derivative of Y with respect to X at the point
m e M. First we follow the integral curve of X through m out to the point
X(m) and evaluate Y there. Then we transfer Yx,(,.) back to M. via the
differential dX_, of the diffeomorphism X_,. In M. we take the difference
of the vectors dX_t(Yx,¢,.)) and Y., divide the difference by t, and then
take the limit as t --, 0. In other words, we consider the smooth M.-valued
function t - dX_t(Yx,¢,.)), and we take its derivative at t -. 0. The result
is a vector m M. which is called the L/e derivative of Y with respect to X at m
and which is denoted by (Lx Y).. Thus we define
(D
dX_,(Yx,(.)) --
(LzY). == lira
I -'0
Y
X

In a similar wry we define the Lie derioatioe of a ffoent fo  with
resct to t orM X, t  in s  we ue   Xt(m)
(m), vide by t, d e e t  tO.. m we d
- X,(z,.)).
Smn of z and of L
dvave L z   exd to  tr elds in  obo w:y.
If T  a tr eId o t (r,:),  (Lz. is  va   -- 0 of
the (M<.,-vu fuon who vue at
_,(, @'-" @ ) @ X,( @... @ :)
(c) Lz: '() '(,   = ri= wi  wi d.
(d) Oa '(, L= -- i(   +   (.
.(, • • •, ,)) - (.=)(,..., ,)
+   .... , ,_, L=,, ,+,..., y,).
( Assumpon  in ().
(,, .... ,)- Z (-- )', ..... , .... ,)
+(- ) [,,], 0 .... ,  ..... , .... , .
<
 We leave sult (a)  an exei. For (b), we  only show
t Lx Y -- [X,for h f ¢ C:(). t m ¢ . Then
=4[

Lie Derioattt 71
Define a real-vaiued function H on a neighborhood of (0,0) in I! by
etfing
(2) H(t,u) -- f(X_,(
Then by (a),
(3)
and (1) becomes
(4)
-- I lt(t,u),
(LxY),(f) " at1 art
To evaluate this derivative, we set
(5) K(t,u,) - l(x,((x,(m)))).
Then ll(t,u) = K(t,u,--t). It follows from the chain rule that
(6)
'// I., a,¢ ] _ a'/ Lo.o.o (
at1 ar--'-- I(o.o) at1 at, I(o,oo) at,
Now, K(t,u,O) i f (Yw(X((m))). Hence
Also K(O,u,s) -- f (X,(Yw(m))). Hence
K I -, Xf(¥,,(m)),
ara I(o.v.o)
(8)
art ara
Part (b) now follows from (4), (6), (7), and (8). One immediate con-
sequence of (b) is that L x Y is a smooth vector field.
For result (c), first consider L x as a mapping of E*(M) into forms
which are not apriori smooth, and observe that the derivation property
follows immediately from adding and subtracting suitable terms before
taking the limit in the definition 2.24(1). Next we check that Lx
commutes with d when applied to functions; that is,
|

and Demal Fovm
(9) (Lx  df)). -- d(Lxf). (f C'(M); m •
Since th sd of (9)  eleen of M, e n only pve
y ve e e et hen app o . bi vtor Y
M.. e At-hd si 
whfX,   ide m a C  [uncon on (--e,e)  W.for
some • > 0 and me neighrh W of m in M (l.(d)).
le-hand s of (9) 
No let Y  an non of Y, to a vor Aeld on . en,
¢o Eerci (c) of Chapter I, d[dt d Y ve nol etensions
fa toother ith (I0) and (l I) impli (9). PitHy, to  tt the
fo Lx is auaHy smoo d to ch at Lx mut ath d
on aH ofE*(M), samply eps  arbitr fo  in II cinates
 in 2.1(2), d compu using on (9), ult (a), d e
tt Lx is a dvaion.
Pot ult (d), oe at bo L x and I(  d + d
derivaons on E*(M) hich commute i d, and th ve the e
e on Junctions. en (d) follos from a simple computation
We sll lea (e)  an eei. e prf is not dictne
simply h to  prsistent enough in unnng the nions. Again,
one chs the identity by rictng to a Io cinae system. T
it Art [or the c   f  dx in a coordina neighrh.
Pinally, ult (0 follos [rom (e) by usg (d) d induon on
Pot the c p  I, e have fm (e)
Applyg (a) and (d)  (12), one
- + +
ch i ult (0 n e c p  I. No su at (0 holds for
p -- I. en (D is obin forp by ain sang ith (e) d applying
(d) d e duon

D/fferenHal Ideal 7
Our objective here is to give a version of the Frobenius theorem in terms of
differential forms, and then to describe E. Cartan's useful method of
obtaining maps by looking for their graphs.
2.26 Deflnition Let  be a p-dimensional C  distribution on M.
A q-form co is said to annihilate  if for each m ¢ M
(l) ¢o,(v: ..... v,) -- 0 whenever v, . . . , v, ¢ (m).
A form co ¢ E*(M) is mid to annihilate  if each of the homogeneous
parts of co annihilates . We let
(2) .'() -- {co  E*(M): co annihilates }.
2.27 Definition A collection ¢oa,..., coo of l-forms on M is called
independent if they form an independent set in M* for each m e M.
Then
(a)
(b)
Proposition Let  be a smooth p-dimensional distribution on M.
.() is an ideal in E*(M).
,() is locally generated by d -- p independen: I-forms. (Tha is,
to each m  M there correspond a neighborhood U of m and a set
of independent l-fornt w 1 ..... w_ on U such that:
(i) lf w  .(), then co [ U belongs to the ideal in E*(U) generated
(ii) If co  E*(M), and tf there is a cover of M by sets U (as above)
such that for each U in the cover, co [ U belongs to the ideal
generated by w: ..... w_,, then co  J().)
(c) If ,  E*(M) is an ideal locally generated by d --. p independent
l-forms, then there exists a unique C ® distribution  of dimension
p on M for which   ().
PRoof Part (a) follows from the definition of ,() and the definition
of multiplication in E*(M).
Let m  M. Since  is smooth and p-dimensional, there exist C ®
vector fields X_+:,..., X defined and spanning  at each point
of a neighborhood of m. This collection can be completed to a collec-
tion X:,..., X of smooth vector fields forming a basis of Mn for
each n in a neighborhood U ofm. Let co: ..... co be the dual l-forms;
that is,
co(X)(n) --  (Kronecker index)
for each n in U. Then co: ..... co_, are the desired l-forms on U.

y are independent smooth l-forms on U. If co e ,(), co J U --
ao,, ^-.. ^ ¢o,,, where ¢ runs over aonempty subsets
{1,... ,d}, and where the a must be identically zero unless
{/,... ,1,)  {1,... ,d-p}  m. =Thm[belongs to the ideal
in E*(U) generated by cos,... , co_,. Conversely, if co is a form such
that for each such U in some covering of M, ¢o ] U belongs to the ideal
in E*(U) generated by cos .... , ¢o_,, then clearly co '(@). This
proves (b).
For part (c), let m  M, and let the independeat l-fortm ot,...
generate .f on a neighborhood U of m. Define (m) to be the subspace
of M, whose annihilator is the subspace of M* spanned by the collection
{co,(m): l -- 1, .... d -- p}. It follows that  is a smoothp-dimensional
distribution on M and that ." -- .'(@). Uniqueness of  follows from
the fact that @ # @s implies '(@) #
2.29 Definition An ideal " c £*(M) is called
if it is dosed under exterior differentiation d; that is,
a differential ideal
2.0 Propoltion A C ® distribution  on M is involutive if and only if
the ideal.() is a differential ideal.
l,tOov Let co be a q-form in 0"(), and let X0,..., X, be smooth
vector fields lying in . Then 2.25(0 together with the involutivene
of  implies that do(X0, .... X,)  0. Hence d¢o f J(), and
'(@) is a differential ideal. Conversely, suppose that '(@) is a differen-
tial ideal. Let Yo and Ys be vector fields lying in @, and let m  M. By
" 2.28(b), there are independent l-forms cot,... ,¢o_, generating
'(@) on a neighborhood U of m. Extend these forms to M by multi-
plying by a C ® function which is I on a neighborhood of m and has
support in U. We shall denote the extended forms similarly by
cot,..., ¢o,. By 2.25(0,
(1) Y0 +
for 1- 1,... ,d--p. The right-hand side of (1) is identically zero
on M since () is a differential ideal and tlnce ¢o  (). Thus
¢o([Ye, Ys])(m) -- 0 for i -- 1, .... d --p. How (m) is the subspace
of M, whose annihilator is the subspace of M* spanned by the collection
{¢o(m): 1 -- 1 .... , d--p}. Thus [Y0, Ys](m)  (m), and isinvolutive.
2.$1 Definition A submanifold (N, lp ) of M is an integral manifold
of an ideal.  E*(M) if for every ¢o , 0(¢o)  0. A connected integral
manifold of an ideal 0" is maximal if its image is not a proper subset of the
image of any other connected integral manifold of the ideal.

Differential Ideals 75
It now follows immediately that we have the following version of the
Frobenins theorem 1.64 in terms of differential ideals.
2.32 Theoel Let . c E*(M) be a differential ideal locally generated
by d--p independent l-forms. Let m ¢ M. Then there exists a unique
maximal, connected, integral manifold of . through m, and this integral
manifold has dimension p.
2.33 We shall now consider a technique which in certain situations enables
one to find a map by looking for its graph as an integral manifold of a
differential ideal. This will have important applications in the theory of
Lie groups in Chapter 3, and also has important applications in such areas
as isometric imbeddings in Riemannian geometry.
Suppose that f: N c--* M  is C ® and that {cot} is some collection of
forms on M. Let r x and rt denote the natural projections of N × M onto
N and M respectively. For each 1, we define a form pt on N × M by setting
(l) p =ffi &rl/f(¢o,) --
Let .f b¢ the ideal in E*(h' × M) generated by the
Now the graph off is the submanifold (h',g) of h' × M where
(2) -
We claim that the graph is an integral manifold of the ideal J. For this,
it suffices to show that Sg(p)-, 0 for each i. Now, 'x °g " id, and
ra o g " f, and thus it follows that
Thus starting with a map f: h'-- M and a collection of forms on M,
we have observed that the graph off is an integral manifold of a certain
ideal of forms on JV × M. Now, suppose that we start with the manifold
M , and suppose that there exists a basis ¢ox ..... ¢o of 1-forms on M,
that is, {(m)} is a basis of M* for each m  M. (Such a basis does not
generally exist, but it does exist in a number of interesting situations for
which the following technique is quite useful.) Suppose also that we have a
manifold h 'c and a collection gx ..... g of l-forms on JV and that we wish
to find a map f: N-- M such that
(3) af(o,,) = ,
for 1 -- 1 .... , d. If the map f exists, then, as we have observed, its graph
will b¢ an integral manifold of a certain ideal of forms. So we attempt to
find the graph from the ideal. We define forms p on JV × M by setting
(4) p = &rx() -
and we let .f b¢ the ideal in E*(JV × M) which they generate. If.f happens
to b¢ a differential ideal, then we can obtain the desired map f (at least
locally) from an integral manifold of.f. That is, one gets atfby looking for

76 7'esor al Dfftk! Forms
its graph as an integral manifold of a suitable differential ideal. So suppose
that .¢ is a differential ideal, and let (no,me) f N × M. Then since .¢ is
locally (in fact globally) generated by d independent l-forms, the Frobenius
theorem guarantees that there is a maxim ¢onnected, integral manifold
I of .¢ of dimenuon c through (no,me). Lei q f i. We clmm that d'l I I, is
1 : 1. For suppose that v f I, and that d.l(v) =ffi 0. Then since p,(v) -- 0,
it follows from (4) that ¢o(dra(v)) =ffi 0 for i -- 1 .... , d; and this impfies
that dn'a(v) =ffi 0 since the ¢o form a basis of l-forms on M. But now if both
da'(v) -- 0 and d'a(v) =, 0, then v =, 0. Hence da' I I, is I : I. Thus
'sll: I-- N is locally a diffeomorphism. So there exist neighborhoods
V of (no,me) in I and U of no such that - [ V: V--* U is a diffeomorphism.
We definer: U--* M by setting
(s) f- "," (,  1l
Thenf(ne) =, me, the graph off is an open submanifold of I, and moreover,
(6) #f(o,,)- l r.r.
For let v  U,, for some n  U. Then by (4),
which proves (6). Thus we have obta/ned the desired map locally. That is,
given no  N and an arbitrary choice of me  M, then under the assumption
that the/deal .f.generated by (4) is a differential ideal, we have found an
open neighborhood U of no and a C ® map f: U-,. M such that f(nd =, me
and such that f(m,) --  [ U. Moreover, there is a unique such map.
More precisely, if me f M, and ff U is any connected open neighborhood
of no in N for which there exists a C ® map f: U-- M such thatf(no) -- me
and such that f(m,) =, , [ U for I =, I .... , d, then there is a unique such
map on U. For letfbe any other such map. Let (Dr, g) and (Dr, g) be the
graphs off and f over U respectively. Thus
(7) [(n) -- (n,f(n)) and g(n) = (n,f(n))
for n  U. Then, according to our remarks near the beginning of this section,
not only is (U,g) an integral manifold of.f through (n0,me) , but so is (V',b).
Now, the subset of U on which g and [ agree is non-empty since it contains
no, and is closed by continuity, and is moreover open. For let n) -- g(n).
Then it follows from the uniqueness of integral manifolds that there exist
suffic/ently small neighborhoods W and I' of n so that
(S) g(W) --/(.
It follows then from (7) that W ,- I' and that
(9) gl w-  l w.

DerenHa/Idea/s 77
Hence, since U is connected, g --  on U, which impfies that f-" f on U.
This proves uniqueness.
In suitable situations one can conclude that -11! is a covering of N;
and then if N is simply connected, one can conclude that there exists a
unique C ® map f: N-* M such that f(no) " mo and such that (3) holds.
In Chapter 3 we shall make several concrete applications of this method for
proving existence and uniqueness of certain maps. This technique was
originally used by E. Cartan for the problem of finding local isometric
imbeddings of Riemannian manifolds in Eucfidean space.
We summarize the results of this section in the following.
2.4 Theorem Let N* and M  be dlfferentiable manifolds, and let
and r o be the canonical projections of N x M onto N and M respectively.
Suppose that there exists a basis (w,: I -, I,..., d) for the l-forms on M.
(a) If f: N  M is C ®, then the graph off is an integral manifold of
the Ideal of forms on N x M generated by
(1) {&rtf(w,) -- &to(w,): l ,= 1,..., d}.
(b) If {q: i ,= 1 , .... d} are l-forms on N, and if the ideal of forms on
IV x M generated by
(2) (&r:(cz) -- 'o(w,): i = 1,..., d}
is a differential ideal, then given no  N and ra o  M there exists a
neighborhood U of no and a C ® raap f: U-- M such that f (ne) =, mo
and such that
(3) ¥(,) = ,1 u (i = 1, .... ).
Moreover, if U is any connected open set containing no for which
there exists a C ® map f: U- M satisfying both f(no) , n and
equation (3), then there exists a unique such map on U.
EXERCISES
1
2
Supply the proofs of 2.2(a)-(e).
(a) Show that homogeneous tensors are generally not decom-
posable.
(b) Show that if dim V  3, then every homogeneous element
in A(I0 is decomposable.
(c) Let dim V > 3. Give an example of an indecomposable
homogeneous element of A(I0.
(d) Let ot be a differential form. Is ot ^   0 9.

3 Supply the proofs of 2.6(a)-(c).
4 Derive the formulas (2), (3), and (4) of 2.10.
$ Prove that LfR Xf whenever f¢ C'(M) and X is a C  vector
field on M.
6 Let X and Y be C" vector fields on M with correspondin local
l-parameter groups X, and Y,. Let m  M, and let
#(t)- _vTx_vT  vT:cv  (m)
for t in (--s,s), for a sufficiently small s.
Prove that
(1) [X,Y]I,f =, lim f(/J(t)) -- f(fl(0))
If/(t) were a smooth curve at t =, 0, then the right-hand side' of
(1) would simply be the effect of the tangent vector to/ at t =, 0
appfied to the function f. However, because of the ,,/,/ is not
generally smooth at t =, 0. Thus the existence of the fimit in (1)
is part of the problem, and the problem asserts that this curve
even though it is not smooth at t =, 0, defines a tangent vector in
M,, in the usual way, and this vector is precisely IX,
7 Prove 2.25(e).
$ Let M be connected, and let -: M × N-,. N be the natural pro-
jection. Prove that a p-form  on M × N is r(a) for some p-form
a on N if and only if i(X) =. 0 and L:ro =, 0 for every vector
field X on M × N for which dr(X(m,n)) ,ffi 0 at each point (re,n)
M×N.
9 Prove that the elements vl,..., v, of the vector space V are linearly
independent if and only if v 1 ^'.. ^ v r # 0.
10 Prove that linearly independent sets (vl ..... vr) and (ws, .... w)
are bases of the same r-dimensional subspace of a vector space V
if and only if vs ^ .-. ^ v r =. cws ^-. • ^  where necessarily
c # 0; and in this case, c =, det A where A =, (a,). and v, =,
11 Let .¢ be an ideal of forms on M locall.y generated by r independent
l-forms. Say .¢ is generated by ol .... , o on U. Then the condi-
tion that .¢ be a differential ideal is equivalent to each of:
(a) d, =, , ^  for some l-forms , (for each such
(r3, , .... ,)).
(b) If  =,  ^..- ^ r, then d =,  ^  for some l-form
(for each such (U, o,..., )).

12
Let V be an n-dimensional vector space, and let I be a lincm" trans-
formation on V. Since A,(V) is one-dimensional, the linear
transformation which I induces on A,(V) is simply multiplication
by a constant. Define the determinant of I to be this constant.
If A is an n × n matrix, let vl .... , v, be a basis of V, and let !
be the linear transformation on V whose matrix with respect to this
basis is A. Then define the determinant of A to be the determinant
of l. Prove that the determinant of A does not depend on the basis
v,,..., v, chosen. Using this definition, derive the standard
properties of determinants. For example, derive the expansion
det A -,  (sgn ')as,¢s • " "
where A -- (a,), sgn - is the sign of the permutation -, and -
runs over all permutations on n letters. Prove also that the deter-
minant of the product of two matrices is the product of their
determinants.
13
Let V be an n-dimensional real inner product space. We extend the
inner product from V to all of A(V) by setting the inner product
of elements which are homogeneous of different degrees equal to
zero, and by setting
(1) <, A-.- A ,, v, A''" A v,> -- det<,,v>
and then extending bilinearly to all of A,(V). Prove that if
ea .... , e, is an orthonormal basis of V, then the corresponding
basis 2.6(1) of A(V) is an orthonormal basis for A(V).
Since A,(V) is one-dimensional, A,(V) - {0} has two components.
An orientation on V is a choice of a component of A,(V) - {0}.
If V is an oriented inner product space, then there is a linear
transformation
(2) .: A(V)- A(V),
called star, which is well-defined by the requirement that for an),
onhonormal basis e,, .... e, of V (in particular, for any re-ordering
of a given basis),
• (I)- e, A'" Ae., ,(e, A''" Ae.)-- I,
(3) ,(e, A-'' A e,) -- e A''' A e,,
where one takes "+" if e, A • "- A e. lies in the component of
A,(V)- {0} determined by the orientation and "--" otherwise.
Observe that
(4) •: A,(V) - A,_,(V).

80 Ten, ors and Diff,,ttal Forms
14
15
16
Prove that on A,(V),
Also prove at for bi v, w ¢ A(, r  pruct is
Oven by
(6) v, - ,(  ,v) - ,(v  ,).
Let V be a real inner product space, as in Exercise 13. Let
Y': A,+I(V)--A,(V) be the adjoint of left exterior multiplication
by  f . That is,
O,(v),) = (v,  ^ )
for v  A.I(V) and w  A,(V). Prove that
v(v) = (-)", • (¢ ^ (.v)).
Let e  V. Prove that the composition
A
A,(V3  A,+,(e) 
of left exterior multiplication by e with itself is an exact sequence;
that is, the image of the first map is the kernel of the second.
Car/u Lemma Let p : d, and let ¢Oa .... , ¢% be l-forms on M 
which are linearly independent pointwise. Let 0a,..., 0, be l-forms
on M such that
Prove that there exist C ® functions A on M with ,4  ,4 such that
O, =  AFo , (| = 1 ..... p).

Lie groups are without doubt the most important special class of differentiable
manifolds. Lie groups are differentiable manifolds which are also groups
and in which the group operations are smooth. Well-known examples
include the general linear group, the unitary group, the orthogonal group,
and the special linear group.
In this chapter we shall set the foundations for the stu.dy of Lie groups.
Of central importance for Lie theory is the relationship between a Lie group
and its Lie algebra of left invariant vector fields. We shall study the corre-
spondence between subgroups and subalgebras and between homomorphisms
of Lie groups and homomorphisms of their Lie algebras. We shall study the
properties of the exponential mapping which is a generalization to arbitrary
Lie groups of the exponentiation of matrices and which provides a key link
between a Lie group and its Lie algebra. We shall investigate the adjoint
representation, shall prove the closed subgroup theorem, and shall consider
basic properties and examples of homogeneous manifolds. Along the way
we shall derive many properties of the classical linear groups.
LIE GROUPS AND THEIR LIE ALGEBRAS
3.1 Definition A Lie group G is a differentiable manifold which is
also endowed with a group structure such that the map G × G --* G defined
by (,-) -, --' is C ®.
Throughout this chapter, G and H will denote Lie groups, and we shall
univemdly use • to denote the identity element of a Lie group.
3.22
$2
Remarks
Let G be a Lie group. Then the map -t-, .-s is C '° since it is the
composition - -, (e,-) -, .-1 of C ® maps. Also, the map (o,-) -, o-
of G × G -- G is C ® since it is the composition (,-) -, (,--1) -, "
of C" maps.

I.,te Groups and Their I.,te Algebras 83
The identity component of a Lie group is itself a Lie group; and the
components of a Lie group are mutually diffcomorphic.
Since we have included second countability in the definition 1.4 of
differentiable manifold, Lie groups for us will always be second countable.
In particular, they can have at most countably many components. It
should be observed, however, that if second countability is dropped
from the definition of Lie group, then connected Lie groups (hence
also those with countably many components) would still be second
countable. We leave the proof of this as an exercise. You will need to
use the fact that a connected Lie group is the union of powers of any
neighborhood of its identity (see 3.18).
3.3 Examples of Lie Groups
(a) The Euclidean space I" is a Lie groupunder vector addition.
(b) The non-zero complex numbers C* form a Lie group under multi-
plication.
(c) The unit circle S t = C* is a Lie group with the multiplication induced
from C.*.
(d) The product G x H of two Lie groups is itself a Lie group with the
product manifold structure and the direct product group Structure;
that is, (, .,,)(, .-,) - (1, .1,).
(e) The n-torus T TM (n an integer > 0) is the Lie group which is the product
of the Lie group St with itself n times.
(f) The manifold Gl(n,) of all n x n non-singular real matrices is a
Lie group under matrix multiplication.
(g) The set of all super-triangular n x n real matrices (all entries below
the diagonal are zero) is a Lie group under matrix multiplication.
(h) Let * denote the non-zero real numbers, and let/C be the product
manifold * x I1. With a group structure on K defined by
(s,t)(sl ,tl) - (ss , st + t),
K becomes a Lie group. This Lie group is the group ofane motions of
., for if we identify the element (s,t) of K with the affine motion
x t-, sx + t, then the multiplication in K is composition of affine
motions.
(i) Let K be the product manifold G/(n,) × '. We define a group
structure on K by setting (A,v)(AI, ) z (AA, Av + ). With this
group structure, K becomes a Lie group. This Lie group is the group of
ane motions of ,", for if we identify the element (A,v) of K with the
arlene motion x t-, Ax +  of ', then the multipfication in K is com-
position of affme motions.

$.4 Definition A Lie algebra g over R is a real vtor spa g together
w/th a b/linear operator [ , ]:  x --,  (cIled the bracket) such that
for'all x,y, z  ,
(a) Ix.y] - -,x].
•
(b) [[x,y]] + [Lv],] + [[,,],y] -- O. (acobi
The importance of the concept of Lie algebra is that there is a special
fin/te dimendonal Lie algebra intimately assodated w/th each Lie group,
and that properties of the. Lie group are reflected in properties of itsLie
algebra. We shall see, for example, that the connected, simply connected
Lie groups are completely determ/ned (up to isomorphism) by the/r Lie
algebras. The stucLy of these Lie groups then reduces in large part to a study
of their Lie algebras.
(a) The vector space of all smooth vector fields on the manifold M forms
a Lie algebra under the Lie bracket operation on vector fields.
(b) Any vector space becomes a Lie algebra if all brackets are set equal to O.
Such a Lie algebra is called abel/an.
(c) The vector space 91(n,t) of all n x n real matrices forms a Lie algebra
(d) A 2-dimendonal vector space with basis x, y becomes a Lie algebra if we
[x.] - ,y] - 0 and Ix.y] - y. ..- - 
and extend bil/nearly.
(e) IR s with the bilinear operation X x Y of the vector cross product is a
Lie algebra.
The proof that the above are Lie algebras is left to the reader as an
3.6 Definitions Let o  G. Left translation by o and right translation by
o are respectively the diffomorphlsms 1, and r. of G defined by
O) I,() -
r.() - 
for all  e G. If gis a subset of G, we denote r.(10 and I.(I0 by I,' and
respectively. A vector field X (not assumed apriori to be smooth) on G is
called/eft/nvaHant if for each  c G, X is l.-related to itseif; that is,
(2) ,a, , x - x , ,.

Lie Groups and Their Lie Algebras
The set of all left invariant vector fields on a Lie group G will be denoted by
the rresponding lowerse orman letter 9.
3.7 Proposition Let 6 be a Lie group and 9 its set of left. invariant
vector fields.
(a) 9 is a real vector space, and the map : 9 - Ge defined by (X) -
X(e) is an isomorphism of 9 with the tangent space Ge to G at the
identity. Consequently, dim 9 " dim G - dim G.
• (b) Left invariant vectorflelds are smooth.
(c) The Lie bracket of to left invarlant vector fields is itself a left
invariant vector fleld.
(d) 9 forms a Lie algebra under the Lie bracket operation on vector
fields.
PxooF That 9 is a real vector space and that  is linear is clear.
injective, for if (X) - (Y), then for each  ¢ G
(1) X(o) - dl,(X(e)) - dl,(Y(e)) - Y(o);
hence X ¥. Moreover,  is surjective; for ifxeG,, we let X(¢)
dlo(X) for each ¢ ¢ . Then (X) -- x, and X is left invariant since
X(.) = dl.oX) = dl. /o(X) - dl.(X())
for all  and - in G. This proves part (a).
For (b), let X¢9, and let f¢C®(G'). We need only show that
Xf ¢ C(O). Now,
Thus we need to show that o - Xeffo !o) is a C ® function on G. We
do this by exhibiting this function as a suitable composition of C ®
maps. Let ¢: G × G - G denote group multiplication, ¢(o,)
And let i x and io t be the maps of G - G × G defined by
io1(.) -
(3)
io'(-) -
Let ¥ be any C ® vector field on O such that ¥(e) -- X(e). Then (0, Y)
is a smooth vector field on O × O, and [(0,Y)(fo q)] o io I is a C ®
function on O. Using part (d) of Exercise 24, Chapter 1, we obtain
[(0, Y)(fo q)] o i?()  (0, Y)°,.,(fo )
= 0o(fo  o i?) + '.(fo  o
- x.(fo  o io,) = x.(fo lo).
Thus o  Xo(fO lo) is a smooth function on G, which proves pan (b).

86 Lit Groutu
Since, by (b), left invariant vector fields are smooth, their Lie brackets
are defined. Now, ifXis l.-related to itself, and ¥is I.-related to itself,
then according to 1.55, [X,Y] is/°-related to [X,Y]. Thus the Lie
bracket of two left invariant vector fields is again a left invariant vector
field. Part (d) is now an immed/ate consequence of 1.45.
3.8 Definition We define the Lie algebra of the Lie group G to be the Lie
algebra 9 of left invariant vector fields on G. Alternatively, we could take
as the Lie algebra of O the tangent space O. at the identity with Lie algebra
structure induced by requiring the vector space isomorphism 3.7(a) of 9
with G, to be an isomorphism of Lie algebras. It will be convenient at times
to consider the Lie algebra from this alternate point of view. In particular,
we shall see (3.10 and 3.37) that the Lie algebras of the classical groups admit
particularly nice interpretations in terms of their tangent spaces at the identity.
3.9 Remarks on Voctor Spaces as Manifolds In 1.5(b) we observe
that any finite dimensional renl vector space I , is, in a natural way, a d/ffer-
entiable manifold. Let {e,) be a besis of I ', and let {r,} be the dual basis.
Now let p ¢ I ,. Then there is a natural identification of the tangent space
I'• to I ' at p with I , itself, given by
Urt l
•
It is easily  that rids identification is indcpandent of tbe beds thorn.
We shall ofn mae  of rids identification in describin tanBent ctors
to • ctor SlU as element of the ector SlU iti. n particular, if
o(t) is a smooth curve in I/, then
(2) 0(t.) - lhn - t0)
,-.,. t -- to
3.10
(a)
Examples of Lie Groups and Their Lie Algebras
The real line  is a Lie group under addition. The left invariant
vector fields are simply the constant vector fields ((d/dr):   ).
The bracket of any two such vector fields is 0.
The General Linear Group. The set gl(n,ll) of all n x n real matrices
is a real vector space of dimension n*. Matrices are added and multi-
plied hy scalars componentwise. As we have already remarked,
Ot(n,) becomes a Lie algebra/fwe set [A,B] .-
The general linear group Gl(n,.) inherits its manifold structure as
the open subset of t(n,) where the determinant function does not
vanish, and is a Lie group under matrix multiplication. Let x u be the
giobal coordinate function on l(n,t) which assigns to each matrix

(c)
its ijth entry. Then if o, ¢ ¢ Gl(n,.),. x,(o "-x) is a rational function
of {x,,(o)} and {x,,(')} with non-zero denominator, which proves that
the map (o,')-,. -x is C'.
Now let g be the Lie algebra of Gl(n,.). Let : 9I(n,)o-,"
be the canonical identification of the tangent space to 9I(n,) at the
identity matrix • with 9I(n,) itself. Thus for v ¢
(1) (v),,- v(x,,).
Then since GI(n,fR)o- 0t(n,lR), we have a natural mp /:
gl(n,IR) dened by
(:z) 0(x)- (x(e)).
We claim that  ts a I. algebra Isomorph/am, and in this way we consider
gl(n,IR) to be the Lie algebra of Gl(n..).  is clearly a vector space
isomorphism. We need only show that
(3) /Z([X.YI) -- U(X),O(
whenever X, Fee. Now,
(4) (x,, o l.X-) - x.(-) = Z
Since l r is a left invt vor eld,
 Z x,()(.
Using (5), we now mpum e # mpnent of ([X, ):
(6) ([x.Y]), = [x.Y],(x,) - x,(Y(x,))- Y,(X(x,))
 Z {(( - (,(x),}
- [(x).(Y)],.
Hen  is a Lie algebra isomosm.
t W  an n-dimensional real vor spa.  End( dote
 of 1 line orato on Y (the t of endomohis of , and
! Aut(Y)c End( denote the subset of non-singular orato
(the automosms). End( is a  vor spa of dimension
d it omes a e gebra if we t

A basis of V determines a diffeomorphism of End(V) with Ill(n,IR)
sending Aut(V) onto ¢71(n,IR). It follows that Aut(V) inherits a
manifold structure as an open subset of End(v) and is a Lie group
under composition. Unde the natu! identification of End(v) with
End(v), -- Aut(V), (where e denotes|dtty h-ansformation on v),
End(v) inherits a Lie algebra structure from the Lie algebra of Aut(v).
This induced Lie algebra structure is precisely the one described in (7).
(d) Let 9l(n,C) denote the set of all n × n complex matrices, and let
Gl(n,C)  9l(n,C) be the subset of non-singular ones. Gl(n,C) is
known as the complex general linear group, gl(n,C) is a 2nS-dimensional
real vector space, with a basis consisting of the matrices d, and
./e-]d, (i,j -- 1 ..... n) where d, is the matrix all of whose entries
are zero except for a ! in the/jth spot; and gl(n,C) forms a Lie algebra
if we set [A,B] -- AB -- BA. GI(n,C.) inherits a manifold structure
as an open subset of gl(n,C) and is a Lie group under matrix multiplica-
tion. With considerations entirely analogous to those in Example (b),
one sees that the natural identification of the Lie algebra of GI(n,C)
with ol(n,C;) is a Lie algebra isomorphism. Thus we may consider
gl(n,C) as the Lie algebra of GI(n,C).
(e) Similarly, in analogy with Example (c), if Visa complex n-dimensional
vector space, and if End(v) denotes the set of complex linear trans-
formations of V, and Aut(V) c End(v) denotes the non.singular ones,
then Aut(V) is a 2nS-dimensional Lie group with Lie algebra End(v).
3.11 Definition A form ¢o on G is called left invariant if
(1) dlo¢o -- ¢o
for each o  G. As in the case ofleft invariant vector fields, it is not necessary
to assume that left invariant forms are smooth, for smoothness is a conse-
quence of the left invariance. We shall denote the vector space of left
invariant p-forms on G by E'tn,(G), and we let
dim 0
(2)
Left invariant l-forms are also known as Maurer-Cartan./'orm.v.
The following proposition is the analog for left invariant forms of 3.7.
The proofs are straightforward, and we leave them to the reader as an
exercise. '

Homomorph£n 89
(d)
Proposition
Left nvarlant forms are smooth.
Env(G) is a subalgebra of the algebra E*(G) of all smooth forms on
and the map co-. co(e) is an algebra isomorphism of Env(G ) onto
A(G). In particular, this map gives a natural isomorphism of Etttv(G)
with G* and hence with ;*. (In this way we shall conlider EIn(G ) as
the dual space of the Lie algebra of G.)
If co is a left invariant l-form and X a left invarlant vector field, then
co(X) is a constant function on G, and this constant is precisely the ect
co has on X when co is considered as an element of the dual space of
as in part (b). (We shall consider co(X) either as a constant function
on G or as the corresponding real number, the particular choice depend-
ing on the context.)
If co  E[tnv(G) and X, Y  g, then it follows from 2.25(0 that
(1) dco( X, Y)
Let {XI ,..., X} be a basis of  with dual basis {col .... , co} for
E,.v(G). Then there exist constants c. such that
Cz) [x,,xJ
(The c. are called the stnctural constants of G with respect to the basis
{X} of .) /'y satisfy
(3) c. + c, -= 0,
The exterior derivatives of the cob are given by the Maurer-Cartan
euations
(4)
l<k
HOMOMORPHISMS
3.13 Definitions A map p: G-*H is a (Lie group) homomorphism
if p is both C ® and a group homomorphism of the abstract groups. We call
p an isomorphism if, in addition, p is a diffeomorphism. An isomorphism
of a Lie group with itself is called an automorphism. If H- Aut(V) for
some vector space V, or if H  Gl(n,C) or Gl(n,.), then a homomorphism
p: G -, H is called a representation of the Lie group G.

If 9 and I are Lie algebras, a map p: 9 "-* I is a (L/e alBebra) honunnor-
phixm ff it is linear and preserves brackets (v[X,Y] -- [V(X),v(Y) ] for all
Y, Y9). If, in addition, p is 1:1 and onto, chen p is an isomorphism.
An isomorphism of 9 with itself is calle,iautomorphlam. If I -- End(F)
for some vector space V, or ffl -- 9l(n,C)o 9l(n, ll), then a homomorphism
P: 9 "* t is called a representation of the l.Je algebra 9.
Let ?: G --* H be a homomorphism. Then zince ? maps the identity of
G to the identity of H, the differential dp of ? is a linear tramformation of
G into H. By means of the natural identificationz of the tangent spaces
at the identities with the Lie algebras, this linear transformation d? of G,
into H, induces a linear transformation of II into I which we shall denote also
by d?. Thus
(1) a,:  --,. ,
where if •  , then dq(X) is the unique left invariant vecto field on H uch
that
(2) ,,(X)() = ,q(X(e)).
3.14 Theorem Let (3 and H be Lie groups with Lie algebras
respectively, and let ?: G -- H be a homomorphism. Then
(a) X and dep(X) are e?.related for each X
(b) d?: II -* t tx a Lie algebra homomorphism.
mtoov Let   d?(X). Then  and X are related. For since ? is a
homomorphism, !(.) o ?  ? o !. ; hence
o) = = aq  (X(e))
= d(l.{o o q)X(e) = d(q o t.)X(e) = dq(X()).
This prov part (a). Now lt X, ¥  ft. Then for part (b), we must
show that
is wrelated to the lift invariant vor field
()
ey 1.55,
In particular,
(3)
t.¢, l'](e) =
But [,Y, IrJ, by Definition 3.13(2), is the unique left invariant vector
field on H whose value at the identity is dq([X, YJ(e)). Thus (2) holds,
and the theorem is proved.
3.15 Effect of Homomorphisms on Left Invarlant Forms Let
q: G--*H be a homomorphism. Then d? pulls left invariant forms on H
back to left invariant forms on G, since whenever o is a left invariant form

Homomorphn 91
Moreover, the mapping dq: £'[v(H)- Ev(G), considered as a mapping of
the dual space of  to the dual space of 9, is precisely the transpose of
dq: 9-" I; that is,
(2) ((co))(x)-co(a(x)) (col,,(H); x9).
Recall that if {co x .... , co} is a basis of E/tnv(H), then there are structural
constants {c,} (see 3.12(4)) such that
(3) dco, --  cv ^
Thus since d and dq commute,
(4) a((co) - :E c, (co) ^
Now let 'x and 's be the canonical projections of G x H onto G and H
respectively. Then the ideal  of forms on ( x H generated by the collection
of independent 1-forms
is a differential ideal; for using (4), we obtain for each i,
(6) a(, (co,) -
+ d.(o) ^ {d,r d(co,) -
which belongs to the ideal . Moreover, observe that the l-forms (5)
generating the differential ideal .k" are themselves left invariant l-forms on
( x H. Thisfollowsimmediatelyfromapplyinglo.totheforms(5),where
(,)eG x H, and from using the fact that .O!o. = !oO. and
' o !o. = leo 's. Note also that the basis {co} of E my(H) is a basis of the
l-forms on H in the sense of 2.33.
To carry out the above construction of a differential ideal on G × H,
it is sufficient to start simply with a homomorphism of the Lie algebras
rather than a homomorphism of the Lie groups. Let G and H be Lie groups,
and let ,p: 9 --* I7 be a homomorphism of their Lie algebras. ,p has a transpose
V*: Ea to(H) E(G), namely,
< (v*(co))(x) = co(v(x)) (co e ,=,(H); x  9).

Let {ca) be a basis of 'v(/O. Then we claim that the ideal " of forms on
G x H generated by the collction of inclopendent l-forms
(s) {,**(,*(o,)) - ,,,(o,:  = .'.. d)
is a differential ideal. This foUows from a computation similar to (6) together
w/th the observation that
(9) (.(o,)) =  ,*(o,) ^ *(o,).
To prove (9), it suffices to prove that   ha  me  on an
bi r ,  of le int vor fl on G:
00)
a (*(c,))(x, ]') - - *(c,)[x, ],] - - o,[x), ]')]
- ao,(x),],))
= c,, ^ o,(x),],))
Here the first and third equalities follow from 3.12(1). Finally, observe
that, in this case also, the forms ($) are left invariant on G × H.
We shall make use of these ideals on G × H in proving existence and
uniqueness of homomorphisms in various situations.
3.16 Theorem Le! G be connec:ed, and let  and  be homomorphLs'ms
of G :o H such :hat :he Lie algebra homomorphLs'ms d and d of g imo [
are identical. Then  , .
I'OOF Since dq -- d, we have
(1)  - : ,*l.,(H)-- l.,().
Thus we have two C** raps q and  of the connected mnifold G into H,
both agreeing at •  G, and both having the same effect of pulling back
a basis of l-forms from H. Thus since the ideal of forms on G × H
geaerated by the l-forms 3.15(5) is a differential ideal, it follows from
2.34(b) that p -- .
3.17 Definitions (H,q) is a Lie subgroup of the Lie group  if
(a) H is a Lie group;
(b) (H,q) is a submanifold of 6;
(c) q: H-- 6 is a group homomorphism.

I. Subgroup 93
(H,?) is called a cloedsubgvoup of G if, in addition, ?(H) is a closed subset
of G.
Let I be a Lie algebra. A subspace [ = I is a Igebra if [X, ]  [
whenever X, ¥  I. A subalgebra b  I clearly forms a Lie algebra under
the bracket induced from I.
Let (H,?) be a Lie subgroup of G, and let [ and I be their respective
Lie algebras. Then d gives an isomorphism of b with the subalgebra
d  (b) of g.
Various theorems assert the existence of unique subgroups satisfying
certain conditions, and as in the case of submanifolds (1.33 is pertinent),
uniqueness needs a little explanaon in view of our definition of subgroup.
We will consider two subgroups (H,) and (Ha,) of {7 equivalent if
there exists a Lie group isomorphism a: H- Ha such that  o a -- .
This is an equivalence relation on the Lie subgroups of O, and uniqueness
for Lie subgroups means uniqueness up to this equivalence. As in the case
of submanifold$ (see 1.33), each equivalence class of Lie subgroups of O
has a unique representative of the form (A,0, where  is a subset of G which
is an abstract subgroup of O and which has a manifold structure (not
necessarily with the relative topology) making  into a Lie group such that
the inclusion i:  -- G yields a submanifold and hence a Lie subgroup of G.
When we wish to consider a subgroup of G in this latter form (i.e. as an actual
subset of G) we usually drop any reference to a mapping and speak simply
of the Lie subgroup  of G, the inclusion map being understood. Also,
we usually identify the lie algebra a of  with di(a) and simply speak of
the Lie algebra of  as a subalgebra of the lie algebra of G. In particular,
we identify a left invariant vector field X on  with the left invariant vector
field di(X) on G with which it is inclusion-related.
We are now going to prove one of the fundamental theorems of Lie
group theory, which asserts that there is a 1 : 1 correspondence between con-
neted Lie subgroups of a Lie group and subalgebrns of its Lie algebra.
But first, we will need the following proposition.
3.18 Proposition Let G be a connected Lie group, and let U be a
neighborhood of e. Then
where U" consists of all n-fold products of elements of U. (We say that U
generates G.)
,Roor Let V be an open subset of U containing e such that V
(where V -x == {o -1: ¢  V}). For example, V == U  U - will do. Let
(2) H

II
Then H is an abstract subgroup of O and is an open subset of O since
o c Himplies oV c H. Thus each coset rood His open in O. Now, H
is the complement in O of the union of all the cosets rood H different
from H itself. Therefore H is also  | subset of O. Since O is
connected, and H is also non-empty,/f must beall of O. This together
with (2) implies (1).
3.19
be a
uch
Theorem Let G be a Lie group with Lie algebra fl, and let   fl
subalgebra. Then there is a unique connected Lie subgroup (tI,) of G
that d¢O() .- .
nu3o We define a distribution  on G by setting
(1) ffi (x(e): xe $)
for each o  O. Let dim - d. en  b  on d.  b
,m ,i  is Ooy ,n by a s ,..., , for . M
over,  is involutive, in ff X d Y am vor fids lg  , th
th  C" funom {a,)  {b,) on G such t -  , d
(2) [X, --  (a,b[XoX].+ a,Xb)X -- bX)X),
wch an is a vor field in  sin  is a subb of .
 (H,)  a mm nn in fold of  tou
• ( I.).  o c (. Sin  is invt und le lafio,
(H, I._,  )    in mold of  ou e. us by 
m, l,-,. (  (. efo ff o  (H) and •  (,
en  x c (. It foIIo at ( is  abs sub, up of
O.  we c indu a up stctu on H  t : H  O is a
homomosm of a Oou. It m oy to chk at H
is a e up, that is, e map : H  H H whe a(o,) ffi o -
is C'. N, e map 0: H  H O nding (o,)  (o)()- is
, d we have e following mmuti diao:
HH
H
Since (H,q) is an integral manifold of an involutive distribution on G,
it follows from 1.62 that g is C ®. Thus (H,q) is a Lie subgroup of G;
and if !) is the Lie algebra of H, clearly dq(l)) = .

 droup 95
For uniqueness, let (K,p) be another connected Lie subgroup of G
with d,p() -= . Then (K,) must also be an integral manifold of 
through e. By the maximality of (H,q), p(K) c q(H), and there is
uniquely determined a map 1 such that q o 1 .= p:
(4) "-,., q
,/is smooth by 1.62, and consequently is an injective Lie group homo-
morphism. Moreover,  is everywhere non-singular; in particular, 
is a diffeomorphism on a neighborhood of the identity, and therefore is
surjective by 3.18. Thus  is a Lie group isomorphism, and the sub-
groups (K,p) and (H,) are equivalent. This proves uniqueness.
Corollary (a) here is a I:1 correspondence between connected Lie
subgroups of a Lie group and ubalgebra of its Lie algebra.
Corollary (b) Let ( It,) be a Lie subgroup of G. Then if If] is a component
of It, (1,  [ 1) is a mammal connected integral manifold of the involutive
distribution on G determined by the subalgebra d(I) of g.
We can restate the content of part of the proof of Theorem 3.19 in terms
of differential ideals on G as follows (cf. Exercise 6):
Corollary (c) Suppose that the ideal l r generated by a collection
{o 1 .... , oe_4} of independent left invariant l-forms on a Lie group G c is a
differential Meal. Then the maximal connected integral manifold I
through e ¢ G is a Lie subgroup of G.
The following theorem hould be compared with the situation for
submanifolds (see 1.33).
3.20 Theorem If an abstract subgroup A of a Lie group G has a manifold
structure (that is, a second countable locally Euclidean topology together with
a differentiable structure) which makes (A,i) into a submanifold of G, where i
is the inclusion map, then it has a unique such manifold structure, and in this
manifold structure, A is a Lie group, and hence (A,0 is a Lie subgroup of G.
etoov We how first that in any such manifold structure, A is a Lie
group. Let  be the distribution on G determined by left translations
of the tangent space to A at the identity. We claim that (A,i) is an
integral manifold of  through the identity e ¢ G.

We caution that this/s not evident and that its proof requ/ras a little
care. The problem is that/f F(r) is a smooth curve in A, and/f o lies
in A, then I. • y(t) is a smooth curve in G which lies in A but apriori
may no longer be smooth in A. Thu,d  that there are eltnents
of (r) which are not tangent vectto A. We must show that this
cannot occur. Let dim A -- k. If for some   A the tangent space
is not contained in (¢), then we can find k -I- I curves which are smooth
in G, lie in A, and have independent tangent vectors at ¢. Translating to
the identity, we have curves yz(t) ..... y(t) which are smooth in G,
lie in A, pass through ¢ at t ffi 0, and have independent tangent vectors
there. Notice that this is not yet a contradiction. For example, there
are two smooth curves in the plane Is with independent tangent
vectors at the origin and lying in the one-dimensional figure-8 submani-
fold (see 1.31). Now consider the map
() (t .... , t.+)  r(t) • • •
This map is non-singular at the origin, thus is a d/fl'eomorphism of a
neighborhood of the origin in I t+z into G, and has its image contained in
A since the individual curves y, lie in A, and A is a subgroup. The map
(5) can be extended to a diffcomorphism of a neighborhood of the origin
in I with a neighborhood U of the identity in G, where n -- dim G.
Composing the inverse of this diffeomorplfism with the inclusion l: A -*
G, we obtain a C ® immersion of the open submanifold U
into I with image containing an open set in t+z  . This now
contradicts the fact that A is second countable and has dimension k.
(See Exercise 6, Chapter I.) Thus A, m D(tr) for each tr¢ A and
consequently (A,i) is an integral manifold of .
It follows that for any ¢ ¢ G, (A,I,) is an integral manifold of
through tr. Thus by 1.59,  is an involutive distribution; and now it
follows from an argument as in 3.19(3) that the map (¢,-)--
of A × A -- A is C ®, which proves that A is a Lie group. Now suppose
that we have two manifold structures on A for which (A,i) is a sub-
manifold. Denote A with these manifold structures by
respectively. Then (A:,i) and (Az,i) are both integral manifolds of
involutive distributions on G; hence 1.62 applied to the commutative
diagram
implies that the identity map id: Ax--As is a
there is a unique such manifold structure on A.
diffcomorphism. Thus

/. Subgroups 97
Hence if a subset of a Lie group can be made into a Lie subgroup under
the inclusion map, it can be made so in one and only one way--the group
structure, topology, and differentiable structure are uniquely determined.
Consequently, it is unambiguous to assert that a subset A of a Lie group G
is a Lie subgroup of G. This means that A is an abstract subgroup of G and
has a manifold structure (and hence a unique one) making A into a Lie group
and (A,i) into a Lie subgroup of G.
We are now in a position to describe exactly in which situation a Lie
subgroup A _ G has the relative topology.
3.21 Theorem Let (H,) be a Lie subgroup of G c. Then  is an Imbed-
ding (that is,  is a homeomorphism of H with (H) in the relative topology)
if and only if (H,) is a closed subgroup of G (that is, (H) is closed in G).
PROOF Assume that q(H) is closed in G. It suffices to prove that there
is some non-empty open set V = Hsuch that q [ Vis a homeomorphlsm
of V into q(H) where q(H) has the relative topology. For then it
follows from the fact that q commutes with left translations (q o
lw  o q) that q is an imbedding on all of H. By 3.19, Corollary (b),
and by 1.60, there exists a cubic-centered coordinate system (U,')
about • ¢ G such that q(H)  U consists of a union (at most countable)
of slices of the form
(1) - m constant for all i
including at least the slice through e. Let C be a closed subset of U
containing e whose image under - is a cube, and let S be the slice of C
given by
(2) '1 '= 0 ..... ' ,= O,
c
Then -(q(H)  S) is a non-empty, closed, countable subset of I c-.
Now, any closed countable subset of Euclidean space must have an
isolated point. Otherwise, such a set with the induced metric would be a

98
complete metric space representable as a countable union of nowhere
dense sets, which is impossible accordin 8 to the Balre category theorem
[27]. Thus there is an isolated slice, call it So, among the collection of
slices (H)  U; and thus -($) is an o .ln subset of H on which 
gives an imbeddin 8 into (H).
Conversely, suppose that p is an imbeddin& Let {o,} be a sequence
ofpoints in p(H) converging to the point o e (. Since p is an imbedding,
there exists a cubic coordinate system (U,) about the identity • e G
such that p(H)  U consists of a singie si/ce--call it $. Choose neigh-
borhoods V c W c U of the identity, culdc relative to -, so that
V -I V  W  U. Since  -- , there is an N so/Sciently large so that
o, eoVforn. N. Therefore Ov-S¢, e Wforn : N. Also -, e q(H).
Thus ov-lo,  $  W and converges to ov-o which therefore
must also lie in S  W. Therefore ov-o  q(H). Hence o  q(H), and
thus q(H) is closed.
COVERIIGS
We shall assume that the reader has some familiarity with the notions of
homotopy, fundamental group, simple connectivity, and covering space.
In Theorem 3.23 we state three of the fundamental facts on covering spaces
which we shall need. The proofs are sketched in Exercise 7 at the end of this
chapter. Details of the proofs and additional facts on covering spaces may
be found, for example, in [26] or [28].
3.22 Definition Before stating Theorem 3.23, we recall a few definitions
and establish some notation. We let ,r(Xx.) denote the fundamental group
of the topological space Xwith base point xe  X. We let st: (X, xe) --,.
denote a mapping of the topological space X into the topological space Y
sending the base point xe in X to the base point Ye in Y, and we denote the
corresponding induced homomorphism of the fundamental groups by
r,: r(X,x,) -- *r(Y, ye).
We shall assume connectedness in the definition of "simply connected."
That is, we shall assume a.ffmply connected space X to be a connected topo-
logical space whose fundamental group is trivial.
A continuous surjection r: X-- Yis called a covering ifXis a connected,
locally pathwise connected (each neighborhood of a point contains a pathwise
connected neighborhood of that point) topological space, and if each point
y  Y has a neighborhood V whose inverse image under r is a disjoint
union of open sets in Xcach homeomorphic with V under . Such neighbor-
hoods V in Y are called even/), covered. If r: X-.- Y is a covering, then Y
is called the base of the covering, and X is called the cover/rig space.
A topological space Y is called semi-locally l.connected if each point
y  Y has a neighborhood U such that each loop based at y and lying in
U is homotoic in Y, through loops based at y, to the constant loop.

Corlng, 99
3,23 Thrm
(a) Let v: (Xco)-- (Y, yo) be a coerin. Let Z be a pathwi, e connected
and local pathwie connected topological space, and let : (Z,)--
(Y, Yo) be a continuo map auch that u..(r(Z,o))  '.('(X,x)).
hen there exists a anue ¢ontinuo map : (Z,o) " (X, xo) uch that
.o,
()
(b) If X i a pathwKce connected, locally pathwie connected, and
locally l-connected topological space, then X ha a sirly connected
(¢) If r: X-- Y i a covering and Y i sirly connected, then v i a homeo-
morpht.n.
3.24 Simply Connd Covering Group Let ,r:   M be a cover-
ing of a differentiable manifold M. Then ./ is automatically a locally
Euclidean, second countable (of. Exercise 8) Hausdorff space, and there is a
unique differentiable structure on ./for which the covering map ,r is ¢® and
non-singular. This differentiable structure is obtained simply by requiring
that the local homeomorphisms obtained from r over evenly covered open
sets be diffeomorphisms.
Now let G be a connected Lie group. By 3.23(b), G has a covering
=: ¢--* G with ¢ simply connected. As we have observed, ¢ has a unique
differentiable structure for which r is C ® and non-singuho. We shall now
show that a group structure can be induced in ¢ making ¢ into a Lie group
and making vr into a Lie group homomorphism. Consider the map
a: ¢ × ¢-- G such that a(O,) -- =(O)=() -. Choose [ c ,r-(e). Since
¢ × ¢ is simply connected, it follows from 3.23(a) that a unique mapping
&: ¢ × ¢ -- ¢ exists such that r o  -- a and such that ([,[) -- [. For
and - in ¢ we define
It follows easily from applications of the uniqueness part of 3.23(a) that
O[ -- O -- O, for the maps O - O[, O - [O, and O - 0 of ¢-*. ¢ all make
the following diagram commute, and they all send [ to [.

100 
Thus since ¢ is simply connected, it follows from the uniqueness in 3.23(a)
that these maps are all identical. Similarly, if follows that 0 "1 -- 0"10 -- ,
and that (o)- O() for all 0, -,  . hus (1) makes G into an
• .. • ,t ', ,;'' .i "-' " -' • •
abstm group; and unce 0 ss smooth,  ts  Lie group. Also, (1) tmphes
that -1)  ,r(.y-I and that ,r() -- ,r(),r('); hence ,r: G -- G is a Lie
group homomorphlsm. Thus we have proved the following.
3.2S Theorem Each connected Lie group  a snply connected cooering
pace wMch is itself a Lie group such that the covering map is a Lie group
].26 Proposition lt O and H be connected Lie groups, and let
O -- H be a bomomorphlsm. Then  is a covering map if and only if
mtoov Suppose first that q is a covering. Then dq ] G, must be injec-
five. Otherwise, since ? is a homomorphlsm, dq would have a non-
trivial kernel at each point of G, and these kernels would form an
involufive distribution on G, whose integral manifolds are collapsed
to points under q. Thus q would nowhere be locally one-to-one,
contradicting the covering property, dq [ G, must also be surjecfive,
for otherwise (G,q) would locally be a proper submanifold of H, again
contradicting the local homeomorphlsm property of a covering. Thus
dq: G, -- H, is an isomorphism.
Conversely, suppose that dq: G, -- H, is an isomorphism. Then q
is everywhere a local diffeomorphism. Thus, by 3.18, ? maps G
onto H since q is a homomorphism and since q(G) contains a neighbor-
hood of the identity in the connected Lie group H. G certainly is
pathwise and locally pathwise connected, so to prove that q is a covering
map, it only remains to show that each point of H is contained in an
evenly covered neighborhood. Let
(1) D -- ker  
Then since q is a local diffeomorphism, D is a discrete normal subgroup
of G, where "discrete" means that if ¢r¢ D, then there is an open set V
in G such that V  D -- or. Since (¢r,-) -, o- is a continuous (in fact
C ) map of G × G -- G, there exists a neighborhood V of e in G such
that
(2) (V -I )')  D -- e.
We claim that q(V) is a neighborhood of e in H, evenly covered by q.
First, q[ Vis 1:1, sinceifcr, - c rand q(cr) -- q0"), then ?(¢") -- e,
whence ¢-.1. _ e by (2), and thus ¢ ffi -. Moreover, dq is an iso-
morphism at each point of G. Thus- q [ V is a. diffeomorphism of V
with the open neighborhood q(V) of e in H. Now, we claim that
(3) - U vs.
f

• .Imply Connected Lie Groups I01
The inclusion = is obvious. To prove , suppose that q(o) e q(V). Then
there exists -  Vsuch that q(-) -- q(o). Hence --o  D and o
proving (3). Finally, V0 s  Vs   if 01  Os  D, for if o  V
then o m - m v/, for some -, v/ V. Then --lv/ D and --v/ V-IV;
thus, by (2), -  v/, which implies that 0 -- 0s. Thus q(V) is an evenly
covered neighborhood of •  H. It follows that q(oV) is an open
neighborhood of q(o) in H, evenly covered by the disjoint union of the
open sets oV for   D. Thus q is a covering.
SIMPLY CONNECTED LIE GROUPS
3.27 Theorem Let (7 and H be Lie groups with Lie algebras g and I
respectively and with G simply connected. Let p: I " IO be a homomorphixm.
hen there exists a unique homomorphi.cm : G -- H such that d , p.
oov Uniqueness was proved in 3.16. •
Let {co} be a basis of the left invariant l-forms on H, and let p* be
the transpose of p (see 3.15(7)). Then according to 3.15, the l-forms
(1) -
on G × H (where =1 and =s are the canonical projections of G × H
onto G and H reslSectively ) are left invariant, and the ideal .f which
they generate is a differential ideal. Thus by 3.19, Corollary (c), the
maximal connected integral manifold I of.f through (e,e) ¢ G × H is a
Lie subgroup of G × H with dimension equal to the dimension of G.
Now, we know from 2.33 that (r [/): I-*- G is nob-singular, and there-
fore by 3.26 is a covering homomorphism. G was assumed to be
simply connected; hence by 3.23(c) and the inverse function theorem,
= [I: I-- G is an isomorphism. We define q: G--,Hby setting
(2) ffi ,,,. (,,1 -1.
Then q is a Lie group homomorphism, and according to 2.33(6),
ffi
Thus dq ffi p, and the theorem is proved.
Corollary If simply connected Lie groups G and H have isomorphic Lie
algebras, then G and H are isomorphic.
There is a theorem [12, p. 199] due to Ado, which we will not prove, that
every Lie algebra has a faithful (1 : 1) representation in gl(n,) for some n.
As a consequence, if g is a Lie algebra, then there is a Lie group, in particular f."
a simply connected one, with Lie algebra g. In view of this we have: 
3.28 Theorem There is a one-to-one correspondence between isomorphism-Z.
classes of Lie algebras and isomorphism classes of simply connected Lie groups. _

I02  Groups
3.29 Deaaitlon
subgroup of G.
A homomorphlsm, q: I1 -- q is called a l.parameter
3.30 Deanidon
LetXe 9. Then
Let G be a Lie group, and let 9 be its Lie algebra.
O) d_.,_.
dr
is a homomorphism of the Lie algebra of  into 9. Since the real line is
simply connected, there exists, by 3.27, a unique l-parameter subgroup
(2) expx:  -- G
such that
(3)
d expx ( -r) --
In other words, t  expx (t) is the unique l-parameter subgroup of  whose
tangent vector at 0 is X(e).
(4)
by setting
(5)
We define the exponential map
exp: 9--G
exp(X) -- expx(l).
The reason for this terminology will become apparent in 3.35(13), where we
show that the exponential map for the general linear group is etually.given
by exponentiation of matrices.

Exponential Map 103
3.31
(0
Theorem Let X belong to the Lie algebra g of the Lie group G. Then
exp(tX) -- expx(t) for each t ¢ .
exp(t + t,)X -- (exp tX)(exp t,X) for all t , ts ¢ .
exp(--tX) -- (exp iX) -I for each t ¢ .
exp: g -," G is C = and dexp: go "* G, is the identity map (with the taual
identifications), so exp gives a diffeomorphlsm of a neighborhood of 0
in g onto a neighborhood of • in G.
i, o expx is the unique integral curve of X which takes the value o at O.
As a particular consequence, left invariant tctor fields are always
complete.
The 1.parameter group of dffeomorphisms X associated tSth the left
inmriant vector fleld X is given by
ooF By 3.14, (d[dr) and dexpx(d[dr), which is X, are expx
related. Thus expx is an integral curve of X and is the unique one for
which expx(0) , e. Since X is left invariant, I, o expx is also an integral
curve of X and is the unique one talcing the value  at 0. Thus part (e)
is proved; and (f) is an immediate consequence of (e). Now define
maps ? and ,p of P into G by setting
(|) ,p(t)  exp, x(t) and ¢p(t) m expx($t)
where $ ¢ I. We have observed that ,p is the unique integral curve
of sX such that ,p(0) "- e. Now,
dcp d , d expx
| •
Thus ¢p also is an integral curve of sX such that ¢p(O) ,= e. By the unique-
ness (1.45(c)) of integril curves, ¢p "- y. Thus
(2) exp,(t) "- expx(st) (s, t
Setting t , 1 and changing $ to t, we obtain part (a). Since expx is a
homomorphism of P into 6, parts (b) and (c) follow immediately
from (a). For part (d), we define a vector field V on G × g by setting
v(,
Then V is a smooth vector field, and according to part (e), the integral
curve of V through (,X) is
t t--* ( exp iX, X),
or in other words, the local l-parameter group of transformations
associated with the vector field V is given by
Vt(o,X) " (o exp tX, X).

104 Lie Groups
In particular, V is complete; hence VI is defined and smooth on all of
G x ft. Now let r: G x fl be the projection onto G. Then
exp X ,= r o VI(e,X). "
Thus we have exhibited exp as the npsifion of C = mappings, so
exp is C ®. That dexp: Io'-* O, is the identity map is immediate, for
tX is a curve in  whose tangent vectOi at t =, 0 is X, and by part (a),
exp tX is a curve in O whose tangent vector at t ,= 0 is X(e).
3.32 Theorem /.et : H-*Gbeahomomorphism,
diagram IS commutative:
H  ,G
.n lhe following
PROOF Let X ¢ b. Then t  p(exp iX) is a smooth curve in G whose
tangent at 0 is d(X(e) This is also a l-parameter subgroup of G
since  is a homomorphism. But t - exp t(&o(X)) is the unique
l-parameter subgroup of G whose tangent at 0 is (&o(X))(e). Thus
(2) (exp tX) -- exp t(d(X)),
whence
(3) q(exp X) -- exp(d(X)).
3.33 Proposition Let (H,) be a Lie subgroup of G, and let X ¢
If X e d(I), then exptX¢ (H) for all t. Conversely,/.fexp tX¢ (H) for
t in some open interval, then X ¢ d(I).
voor If Xedq(IT), then exptX¢ p(H) for all t, according to 3.32.
On the other hand, if exp tX ¢ (H) for t in some interval I, then for
t ¢ I, the map t - exp tX can be expressed as a composition q o  where
• , according to 1.62, is a smooth map of I into H. Let to ¢ I, and let
• be the left invariant vector field on H determined by (to). Then
- X.
3.34 Theorem Let A be an abstract subgroup of a Lie group O, and let a
be a subspace of . Let U be a neighborhood of 0 tn  dffeomorphic under the
exponential map w[th a neighborhood V of the Identity in G. Suppose that
(1) exp(U  a) -- A  V.
Then A with the relative topology IS a Lie subgroup of G, a is a subalgebra of
g, and a is the Lie algebra of A.

Expanential Map 105
voor We need only how thatin the tlative topology, the subgroup
A has a differenfiable g, xucture such that (A,i) is a submanifold of G,
where i is the inclusion map. For then, according to 3.20, with this
manifold structure (and no other) A is a Lie subgroup of G, and it
follows from 3.33 that the Lie algebra of A is a. Let
¢p--exp[Ua:Ua -- ACV.
Then the desired differentiable structure on A is the maximal collection
of smoothly overlapping coordinate systems containing the collection
$.$5 Example We shall see that the exponential map
(1) exp: 0l(n,C) -* GI(n,C)
for the complex general linear group is given by exponentiation of matrices.
We shall now let I rather than • denote the identity matrix in Gl(n,C). Let
A
¢2) -I+A+. +-..+. +'-.
for A  gl(n,C). "to see that this makes sense, that is, to see that the right-
hand side of (2) converges, observe that in fact the fight-hand side of (2)
converges uniformly for A in a bounded region of gl(n,C). For given a
bounded region [1 of Ol(n,C), there is a  > 0 such that for any matrix A
in this region, Ixt(A)l < t for each component xa(A) of the matrix A.
It follows easily by induction that Ix,(A')l < nC-a'' . Thus, by the Weier-.
strass M-test, each of the series
y x,(t') (1 < i < n; 1 < k < n)
(3) ,--o
converges uniformly for A in . Thus the right-hand side of (2) converges
uniformly for A in ft.
Now let S(A) be thejth partial sum of the series (2), that is,
A t A. 
(4) S(a) = I + a + . +"" + i,. ,
and let B  oI(n,C). Then since C  BC is a continuous map of OI(n,C)
into itself, it follows that
In particular, if B  GI(n,C), then B()i_..m S(A))B -x
lira
BS(A)B --1 ,
from
which it follows that
(6) se s- - eBbS-"

10  Groups
There exists a B e 71(n,C) such that BAB -I is super-triangular; that is,
all entries below the diagonal are zero. (Simply take B to be the inverse of
the matrix whose columns vl .... , v= are determined as follows: Let !
be the linear transformatio of C" whose matrix with respect to the canonical
basis is A, let vt be an eigenvector of I, and inductively let v,+t be an eigen-
vector of ., o i
spanned by v ..... v and where . s projection of C" -- V  W onto W, .)
If the diagonal entries of BAB - are 1,..., ,, then • a-I is also super-
triangular, with diagonal entries el,..., e -. In particular, det eaS-x# 0,
so it follows from (6) that
(7) •  c G/(n,C) for each A c gt(n,C).
Since the trace of a matrix is the sum of its eigenvalues, and the determinant
is the product of its eigenvalues, we have also shown that
(8) det
Next we shall prove that
(9) ¢+s. e.eS if AB -- BA.
By definition, e +n -- lira S(A + B). It follows from the fact that matrix
multiplication gives a continuous map of gl(n,C) x l(n,C) into l(n,C)
that e'/e  -- lira S(A). S(B). So to prove (9), it suffices tb show that
(10) lira {$(A)$(B) -- $(A + S)} -- 0.
Now,
(11) $(A)S(B) -- S(A + B) -- . l!k---'!
where the sum is over all integers I and k for which 1 : 1 : A 1 : k : j,
and j + 1 < i + k < 2j. If/ > 1 is an upper bound for the absolute value
of each of the entries of A and B, then it follows that each entry of the right-
hand side of (11) is bounded in absolute value by
(12) , n+x#'t < (n#)t(Jt) "
!i!k! [1/2]!
where [j/2] is the greatest intelier.less than or equal toil2. The etimate on
the right-hand side of (12) goes io zero asj-, , which proves (10) and hence
proves (9).
Consider the map t - •  Of R into G/(n,C). It is smooth since the real
and imaginary components of leach entry of •  are power series in ! with
infinite radii of conversence. Iis taagent vector at 0 is A (simply differentiate
the power series term by term i, and this map is a homomorphism by (9).

F.xponent Map 107
Thus t  • t is the unique l-parameter subgroup of Gi(n,C) whose tangent
vector at 0 is A. So the exponential map (1) for Gl(n,C) is given by exponen-
tiation of matrices:
03) e  l,(A) - (A
IfA  9I(n,ll), then e "/ Gi(n,l), and A --* e "i is the exponential map for the
real general linear group. Similarly, the exponential map
(14) exp: End(F) --* Aut(F),
where V is a real or complex vector space, is given by exponentiation of
endomorphisms. If l  End(V), then
(Is) - e'- 1 + l + +'" + +""
where P means "i composed with itselfj times," and where 1 is the identity
transformation.
$.$6 Remarks If q: O-*Aut(F) is a representation, and Z¢0, then
it follows from 3.35(15) and 3.32 that
(1) q(exp X) -- 1 + dq(X) + (dq(X))S +'",
2!
and it follows from 3.32 and 3.9(2) that
lim,.e qexp tX)t -- l= atl'--dl-o(qexp tX)).
(2)
dX)
$.37 Subgroups of Gl(n,C) Using 3.34 and 3.35, we shall now obtain
some of the classical Lie subgroups of Gl(n,C) by exponentiating subalgebras
of 0l(n,C). We already have one example, namely, Gl(n,) is a closed Lie
subgroup of Gl(n,C) with Lie algebra 0l(n,)  0l(n,C). Let A ¢ 0l(n,C).
Then the transpose of A is the matrix A t such that (At)j -- A,; and the
complex conjugate of A is the matrix  whose./jth entry (.), is the complex
conjugate A' of A,. And n, of course, is an integer  1.
(a) Unitary group U(n)
(b) Special linear group Sl(n,C) :: {A ¢ Gi(n,C): det A -- 1}.
(c) Complex orthogonai group O(n,C) -- {A ¢ Gl(n,C): A - -- At}.
Each of these is an abstract subgroup and a closed subset of Gl(n,C).
We shall use 3.34 to show that they are Lie subgroups. Their Lie algebras
will bc
(a') Skew-hermitian matrices u(n) -- {A  0l(n,C): A" + A  -- 0}.
(b') Matrices of trace 0 $I(n,C) = {A  0I(n,C): trace A -- 0}.
(c') Skew-xymmetric matrices o(n,C) -- {A  0l(n,C): A + A t = 0}.

Each of these examples is a subalsebra of gt(n,C). To apply 3.34, let
Ube a neighborhood of 0 in gl(n,C), difl'eomorphic under the exponential
map with a neighborhood g of the identity in Gl(n,C). We can assume, in
addition, that ira • U, then , A t, and --A belong to U, and trace AI < 2..
For let W be a neighborhood of 0 in iI(nC) that is small enough for
the exponential map to be a diffeomorphism, and also small enough for
the trace condition to be satisfied, and thenlet U -- W C ' C W* C (-- W).
We shall also assume that exp(U c l(n,)) -- G/(n,) C g.
If A • U C u(n), then (),. e r' , e-; hence ()*e  , e--e , ,
e* -- I, which implies that •  • U(n) c V. Conversely, suppose that
A • U and thaf  • U(n) C V. Then e "- -- (e-/)-I -- (X), _ el,, which
implies that --A -- X' since --A and Xt also belong to U and since the
exponential map is 1 : I on U. Thus A • U c u(n). It follows from 3.34
that U(n) is a closed Lie subgroup of G/(n,C) with Lie algebra u(n)., A
similar argument shows that O(n,C) is a closed Lie subgroup of Gl(n,C)
with Lie algebra o(n,C).
If A • $[(n,C), then by 3.35(8), det •  -- I ; hence e  • $1(n,C). Con-
versely, if det e  , 1, then according to 3.35(8), trace A -- (2.i)j for some
integer j. If in addition A • U, then trace A -- 0. Thus 3.34 implies that
$/(n,C) is a closed Lie subgroup of Gi(n,C) with Lie algebra $[(n,C).
It follows immediatety from 3.34 and from considerations as given above
that the special unitary group SU(n), which is by definition U(n) C $1(n,C),
is a closed Lie subgroup of Gl(n,C) with Lie algebra su(n) the suhalgebra
of l(n,C) consisting of skew-hermitian matrices of trace 0; the real special
linear group $1(n,), which is by definition $1(n,C) c Gl(n,), is a closed
Lie subgroup of Gl(n,) with Lie algebra $[(n,) the real matrices of trace
0; the orthogonai group O(n), which is by definition I(n) C Gl(n,), is
a closed Lie subgroup of Gi(n,) with Lie algebra o(n) the real skew-
symmetric matrices; and the special orthogonai group SO(n), which is by
definition O(n) c $1(n,), is a closed Lie subgroup of Gl(n,) also with
Lie algebra o(n) the real skew-symmetric matrices.
Each of these examples is a closed subgroup of either Gl(n, ) or Gi(n, C),
and it follows from either 3.34 or 3.21 that in each case the Lie topology is
the relative topology.
The unitary group is compact since not only is it closed, but also it is
bounded in G/(n,C), for AtX -- I implies that  [A,[ s , I for each i, which
implies that [A,[s <: 1. It follows that SU(n, O(n), and SO(n) are also
The dimensions of these Lie groups are easily computed from their
Lie algebras. U(n) has dimension nt; $1(n,C) has dimension 2n t -- 2;
O(n,C) has dimension n(n -- I); SU(n) has dimension n t -- 1; $1(n,.)
has dimension n t -- I; O(n) and SO(n) both have dimension n(n- 1)/2.

Continuous Homomorphisms 109
CONTINUOUS HOMOMORPHISMS
3.38 Theorem Let : --* G be a continuous twmomorphism of the
real line  nto the Lie group G. Then  is C "°.
p]xoo It sutlices to prove that q is C ® on a neighborhood of 0, for
then it follows from composition with suitable left translations that
q is C ® everywhere. Let V be a neighborhood of • • G di/Teomorphic
with a neighborhood D r of 0 • I under the exponential map. We can
assume that D r is starlike, that is, tX • D r for 0 <: t <: 1 whenever
X • D r, and we let
U' - {X/2: X •
Choose te > 0 small enough so that q(t) • exp(U') for Itl <: re. Let
n be a positive integer. Then there are uniquely determined elements
X and Y of D r' such that exp X ,= (to/n) and exp Y- q(te). We
claim that nX -- Y. Since
exp(nX) ,= ?(to) -- exp(Y),
and since exp is injective on D r', we need only show that nX¢ U'. Now,
X• U', so let 1 <:j < n, and assume thatjX• U'. We will prove that
(j+ I)X¢ U'. Now,(j+ I)X• U, andexp((j+ I)X) ,= q((j + l)to/n)
which by assumption belongs to exp(U'). Since exp is injective on U,
it follows that (j+ I)X• U'. Hence nX• U' and nXm Y, as claimed.
Now, let m be an integer with 0 < Iml <2 n. If m is positive, then
?(into/n) ,= ?(to/n)" ,= exp(Y/n)"* ,= exp(m Y/n). If m is negative, then
also ?(mteln) ,= ?((-m)toln) -x  exp((-m)YIn) -x "= exp(mY/ta). It
follows by continuity that ?(t) ,= exp(tY/te) for Itl < to. Thus ? is C ®.
3.39 Theorem Let : H--, G be a continuous homomorphism of Lie
groups. Then  i$ C .
vary t H  of dimension d, and let X ..... X  a basis of b.
e map u:   H defined by
(!) (t ..... t)  (exp tX)'." (exp tX)
is C , and is non-singular at 0 ¢  by 3.31(d);  the is a neibor-
hood V of 0 ¢  diffeomorphic under • with a neiborho U of •
in H. Now, t  (exp tX) is a continuous homomohism of
G, and so is C  by 3.38. us   • is C , and efo  ] U, which
can  exposed as ()-]U, is C
1. o  o l-x I U,  is C  on all of H.
.40 Definition A topologlcol group G is an abstract group G which
has a tolo such that the map(,)  - of G x G  G is continuous.

110 Lie Groups
3.41 Corollary to Theorem 3.39 1 second countable locally Euclidean
topological group can have at most one differentlable structure making it into
a Lie group.
PROOF The identity map would give a diffeomorphism of any two. such
differentiable structures.
One of the outstanding problems in the theory of Lie groups was that of
deciding whether every connected locally Euclidean topological group has
a differentiable structure which makes it into a Lie group. The problem was
posed by Hilhert in his famous address to the International Congress of
Mathematics in 1900, and was solved with an affirmative answer by Gleason
together with Montgomery and Zippen in 1952; see [20].
We have dealt exclusively with the C  structure on Lie groups. It can
be shown [23] that the C ® structure on a Lie group contains an analytic
structure, that is, a collection ofcoordinate systems which overlap analytically
(the compositions % o %- of coordinate maps being locally represented
by convergent power series). In this context, the analog of 3.39 would state
that every continuous homomorphism of Lie groups is analytic, from which
it follows that the C  structure on .a Lie group contains a unique analytic
structure.
CLOSED SUBGROUPS
3.42 Theorem Let G be a Lie group, and let A be a closed abstract
subgroup of G. Then A has a unique rr, anOold ,tructure which makes A into
a Lie subgroup of G.
According to 3.21, the topology in this manifold structure on A must he
the relative topology.
PROOF Uniqueness has been proved in 3.20. Let
(!) a -- {Xcg: exp tXcA for all t¢ I}.
The idea of the proof is to show that a is a subspace of g, and to apply
3.34. It is clear from the definition (1) that if X ¢ a, then also tX ¢ a for
any t ¢ I. Now, let X, Y ¢ a. Suppose that
(2) lim xp t_. X exp t_ ,= exp(t(X + Y)).
Since A is closed, the left-hand side of (2) belongs to A. Thus (2)
implies that (X + Y) • a. Assuming (2) for the moment, we have proved
that a is a subspace of I and shall use this to complete the proof of the
theorem. We shall return to the proof of (2) in a separate lemma.

Closed Subgroups 111
The theorem will follow from 3.34 once we have shown that there
exists a neighborhood U of 0 in g, diffeomorphic under the exponential
map with a neighborhood V of • in G such that
(3) exp(U t a) = V t A.
Suppose, on the contrary, that no such U exists. Then there is a
neighborhood W of 0 in a and a sequence {or}  A such that o t-- •
in G and
(4) ot  exp(I¢).
Letb be any complementary subspace to e in g. It follows from 3.31(d)
that the map ot: e × b --* G defined by o(X, Y) -- exp X exp Y is C ®
and non-singular at (0,0). Thus there are neighborhoods IVo  W of
0 in a and lVb of 0 in b such that ot ] IV o × We is a diffeomorphism of
IV o × We with a neighborhood 117 of • in G. If we can choose IVe
small enough that
(5) A t exp(We- {0}) ,= ,
then we can reach a contradiction as follows. For k large enough,
at  117, and thus at ,= exp Xt exp Yt for Xt  IVo and Yt  IVe, where
Yt # 0 by (4). But ot  A, as also exp Xt  A; so exp Yt  A, contra-
dicting (5).
Finally, we show "that lVe can be chosen satisfying (5). Again we
argue by contradiction. Suppose that there is a sequence { Y}  We
such that Y # 0, Y -- 0, and exp Y  A. Then there is a subsequence
{ Y} and a sequence {t} of positive real numbers with t --- 0 such that
the sequence { Y/t} converges to some Y # 0 in We (choose a norm
on the vector space We, and let t be the norm of Y). For t > 0,
let n(t) be the largest integer less than or equal to (tits). Then
--- 1 < n(0 <-,
so lira tn(t) *= t. Thus
exp tY -- exp(lim n(t)Y) ,= lira (exp
which belongs to A. Since exp(--tY) = (exp tY) -1, then exp tY A
for all real numbers t, which implies that Y ¢ a. This contradicts the
fact that Y is a non-zero element of We.
To complete the proof we need the following lemma,, which clearly
implies (2).

|12 LIO roup
Imma let 6 be a Lie group with Lie algebra g. If X, Y • , then for t
(6) exp tXexp tY -- oxp(t(X + Y) + O(tt)),
where O(t t) denotes a I)-valued C "° function of t such that (l]tt)O(t s) is
bounded at t ,- O.
IooF For t small enough, there is a C  curve Z(t) in $ such that
(7) exp tX exp t Y -- exp Z(t).
Since the tangent vector to the curve
t - exp tX exp t Y
at t -- 0 is X(e) + Y(e) (se Exercise 11), it follows that the tangent
vector to Z(t) at t -- 0 is X + Y, so the Taylor expansion with integral
remainder of Z(t) about t -- 0 has the form
(s) z(t) - t(x + + o(:),
where O(t t) is a C  g-valued function of t such that (l/tl)O(t t) is
bounded at t. -- 0. From (7) and (8) we obtain (6).
$.45 Theorem Let ,: G- K be a homomorpht.n of Lie gro,.
,4 -- ker , and a -- ker d, then ,4 is a cioaed Lie ubgroup of G with Lie
algebra a.
OOF A is a closed abstract subgroup of G, hence, by 3.42, is a Lie
subgroup of G. If X • i, then aceording to 3.33 (with H -- A and 
the inclusion map), X belongs to the Lie algebra of A if and only if
exp tX•A for all t • , and this occurs if and only if(exp iX) -- •
for all t • II. By 3.32, this latter condition is equivalent to having
exp t d,(Z) -- • for all t • , and this occurs ifand only if d,(X) ,= 0
or X• a.
THE ADJ'OINT REPRESENTATION
3.44 Defln/tions Let M be a manifold, and let G be a Lie group. A
C  map p: G x M - M such that
(1) p(a',m) -- p(a,.u0",m)), p(e,m) -- m
for all , " ¢ G and m • M is called an action of G on M on the left. If
p: G x M - M is an action of G on M on the left, then for a fixed  • G
the map m ,--, p(o,m) is a diffeomorphism of M which we shall denote by po.
Similarly, a C" map p: M x G - M such that
(2) - m
for all , " • G and m • M is called an action of G on M on the right.

Th Adjo¢nt Rprscntat¢on 113
:3.45
left.
¢  G. Then the map
defined by
()
is a representation of G.
Theorem Let p: G x M -- M b an action of G on M on the
Assume that mo e M is a fixed point, that is, po(mo) -- mo for each
: G -* Aut(M)
v(¢)
is a C ® map of G into M,o, or equivalently that (4) is a C ® map of G
into T(M). But (4) is exactly the composition of C ® maps
¢  -() × T(M)-* T( × M)-* T(M)
in which the first map sends  , ((o,0),(me,Ve)), the second map is the
canonical diffeomorphism of T(G) × T(M) with T(G × M), and the
third map is d r. Thus ,p is C ®.
3.46 The Adjoint Representation A Lie group G acts on itself on
the left by inner automorphisms:
(1) a: G x G---G, a(o,') '= ¢r - '= ao().
The identity is a fixed point of this action. Hence, by 3.45, the map
(2) o'daolG°----g
is a representation of G into Aut(g). This is called the adjoint representation
and is denoted by
(3) Ad: G -* Aut(g).
then
(3)   =(dr,(v0))
is a C ® function on G. For (3), it suffices to prove that
(4)   dro(to)
],xoov , is a homomorphism for
(3) v() = dr,, I M., = d(r, o r,) I M.. = v() o v().
It remains only to prove that y is C ®. For this, it suffices to prove that
y composed with an arbitrary coordinate function on Aut(M..) is C .
Now, one gets a coordinate system on Aut(M.) by choosing a basis
for M.o and then by using this basis to identify Aut(M.) with non-
singular matrices. One gets the matrix associated with an element of
Aut(M,.o) by applying this element to the basis of M,o and then applying
the dual basis. So it suffices to prove that if v0  M. and if =  M*o,

We let the differential of the. adjoint representation be denoted by ad,
(4) d(Ad)
and we denote Ad() by Ado and ad(X) b.y ad x . Thus by 3.32 we have a
commutative diagram
(5)
G Ad  Aut(g)
exPl a d l 'xp
g - End(g)
Also applying 3.32 to the automorphlsm a of G, we obtain the commutative
diagram
(6)
In other words,
(7) exp tAdo(X) -- (exp tX)o "-1.
In the special case in which G -- Aut(V), the above diagrams become
(8)
Aut(V), Ad • Aut(End V)
End(v) ...... ad  End(End V)
Aut(V), ) ) Aut( i f)
End(v) ' Ad, )- End(v)
where B  Aut(v). If in addition C  End(v), then
(9) AdB(C)
For from 3.36(2), using 3.35(15) and 3.35(6), we obtain
d l,.o(a(ex p d[
Ade(C) "= t tC)) .=  (B o e *c. B-)
dt

Th Adjoint Representation 115
Similarly, in the case in which G -- G/(n,R)(or G/(n,C)) and B
and C e gl(n,l), then
(10) AdB(C) -- BCB -1.
$.47 Propmition
X, Y  . Then
let G be a Lie group with Lie algebra g, and let
PltOOF
(2)
(1) adxY ,ffi IX, Y].
By $.$6(2),
"dxY'(t[,.oAd(exptX))Y
[..o A d [
Thus if X, denote,  usual, e l-mmeter oup of diffeomohisms
assiat wi X, then
d [ d(re.(_,x))(d(leffi  ,x)(y(e)))
(3) adxY(e) =  
 dt 
d  d(X_t)(Y,(,))
- ffi
The last equafity follows from 2.25(b). Sin adz Y and
both left invaant, (I) is  immia nuence of (3).
3.48 Theorem Let A  G be a connected Lie ubgroup of a connected
Lie group G. Then A is a normal mbgroup of G if and only if the Lie algebra
a of A is an ideal in 9.
Ptoo Assume that a is an ideal in 9.
 -- exp X. Then
(1) (exp Y)-I ffi exp Ad,(Y)
ffi exp((exp adx)(Y))
Let Y ¢ a, let Z ¢ , and let
(3.46(7))
(3.46(5))
(ad)2X )
• ffiexp Y+[X,Y]+(Y)+"" •
(3.35(15) and 3.47)

116 Lit
Under the assumption that a is an ideal in I1, the eries in the last term
in (1) converges to an element of a, o that
(2) o(exp Y)o- e A.
Now, by 3.18 and $.$1(d), A is enerated by elements ofthe form exp Y,
and G is enerated by elements of the form exp X. This, tother with
(2), implies that A is s normal subgr6upof G.
Conversely, assume that A is normal in G. Let s and t be real numbers,
let Yea and Xe g, and let o -- exp iX. Then, according to (1),
($) o(exp sY)o -1 m exp Ado(s Y) -- exp s{(exp ad,x)(Y)}.
Since A is normal, o(expsY)¢ -s e A; so it follows from ($) and
3.33 that (exp ad,x)(Y) e a for all t e I1. Now
(4) (expadtxXY) . (exp t(adx)XY )
t s
.. y + ttx, yl + [x, tx, yl] +--.,
which is s smooth curve in e whose tansent vector at t -- 0 is IX, Y].
Thus IX, Y] e a, and e is an ideal in 8.
$.49 Delinittom
(a) Center of 0 -- (X ¢ 0: [X, Y] , 0 for all Y ¢ g}.
(b) Center of G , {¢ e G: ¢y" , ,'¢ for all r ¢ G}.
3.50 Theorem Let G be a connected Lie group. Then the center of G
is the kernel of the adjoint representation.
tntool, Let o belong to the center of G, and let X ¢ 0. Then
(1) exp tX m o(exp tX)o -a . exp tAdo(X')
for all t e I1. Thus X . Ado(X); so  e ker(Ad). Conversely,
let  e ker(Ad). Then (1) again holds, so  commutes with every element
in a neighborhood of • in G. Since G is connected, o commutes with
every element of G.. Thus  lies in the center of G.
Corollary (a) Let G be a connected Lie group. Then the center of G is a
closed Lie subgroup of G with Lie algebra the center of O.
ROoF This corollary is an immediate consequence of 3.50 and 3.43.
Corollary (b) .4 connected Lie group G is abelian if and only if its Lie
algebra 0 is abellan.
Proposition Let X, Y belong to the Lie algebra tIofa Lie group
IX, Y] . 0  exp(X + Y). exp X exp Y.

Automorphtn & Derivation of Bilinear Operation ! 17
oo The subspace a of fl spanned by X and Y is a subalgebra of fl,
and the corresponding connected Lie subgroup A of G is abelian. Let
z(t) ,,= exp tX exp t Y.
Then 0 is a C  map of IR into G and is a homomorphism since A is
abelian. The tangent vector to 0 at 0 is X(e) + Y(e) (see Exercise 11),
so that
exp tX exp t Y -- exp t(X + Y)
for all t.
$.52 Remark Recall that in 3.27 we stated Ado's theorem which
asserts that any Lie algebra g has a faithful representation in some 0l(n,R)
from which it follows that there is a Lie group with Lie algebra g. If g has
trivial center, we can get such a faithful representation from the adjoint
representation. Simply define ad: --* Edd(o) by adx(Y), IX, Y], and
check that this is a Lie algebra homomorphism. If  has trivial center, ad
is one-to-one; so we have a faithful representation of 0 in End(0). This
together with 3.19 gives a simple proof that every Lie algebra with trivial
center is the Lie algebra of some Lie group.
AUTOMORPHISMS AND DERIVATIONS OF BILINEAR
----OPERATIONS AND FORMS .
3.53 Definitions Let V be a finite dimensional real or complex vector
space. A bilinear operation on V is a linear map *p: V® V-* V where the
tensor product is taken over the real (resp. complex) numbers when V is a
real (resp. complex) vector space. We shall use the notation
(v ® w) - {v,w).
(a) We let
A.(V) , {0  Aut(V): 0{v,w} , {0(v),0(w)} for all v, w ¢ V}.
The elements of A,,(V) are the hutomorphisms of V which preserve
the bilinear operation
(b) We let
b , {i ¢ End(V): l{v,w} , {i(v),w} + {v,l(w)} for all v, w  V}.
Elements of b,. are called derivations of the bilinear operation
3.54 Theorem AT(V ) is a closed Lie subgroup of Aut(v) with Lie
algebra b.

118 /. ¢Troup
moo It is easy to check that b, is a subalgebra of End(V) and that
A(V) is a closed abstract subgroup of Aut(V). According to Theorem
3.42, A(V) is a  Lie ubgroup of Aut(¥). Let a be the Lie algebra
of A(V). We need only show that a-- b.
If/ca, then exptlA(V), so that
(1) (exp t/){v,w} . {(exp.(v), (exp t0(w)}.
Both sides of (1) are smooth curves in g. Taking their derivatives at
t . 0, we obtain
(2)
which proves that I c b. (Observe that a curve ?(t) ® (t) in g ® g
has derivative (or tangent vector) at t- 0 equal to q(0)® (0)+
v(0) ® (0).)
Conversely, suppose that I c b,. To prove that I c a, we need bnly
prove that exp tl A(V) for all t. We let ! ® I denote the endomorphism
of g ® g defined by
The fact that ! c b means that
(3)
for all v and w in V; and this can be expressed as
(4) o = o(® I + I ®0.
It follows from (4) that
() -o. .(l® l + t ®0%
so that
(6) e
Now, (! ® 1) o (1 ®/) -- (1 ®/) • (I ® 1), so wc can apply 3.35(9)
to the right-hand term in (6). Then, by the fact that • toes -- e a ® 1,
equation (6) becomes
(7)
Thm
(exp t0{v,w} - {(exp t0(v), (exp t0(w)}
for all
Thus a -- b, and the theorem is proved..
3.55 Definitions Again we let V be a finite dimensional vector space
over a field F where F is either  or C. A bilinearform : on V is a Hncar
map B: V® V-* F. We shall use the notation B(v ® w) -- (v,w).

Automorphisms & Derivation of Bilinear Operations 119
(a) We let
An(V) ,= {oc  Aut(v): (v,w) , (oc(v)#(w)) for all v, w  V}.
The elements of An(V) are the automorphisms of V which preserve
the bilinear fom B.
(b) We let
bn = End(v): (l(v),w) + (v,l(w)) -- 0 rot w V).
Elements of bn are called derivations of the bilinear form B.
$.56 Theorem An(V) is a closed Lie subgroup of Aut(V) with Lie
algebra bn.
The proof is similar to the poof of Theorem 3.54, and we leave it to the
reader as an exercise.
(b)
Applications of $.54 and $.56
Let V be a real vector space with an inner product B. According to
Theorem 3.56, the automorphisms of V which preserve the inner
product form a closed Lie subgroup An(V )  Aut(V) whose Lie
algebra is the Lie subalgebra bn  End(V) of derivations of the inner
product B. Choose an orthonormal basis for V. Now consider the
Lie group isomorphism of Aut(V) with Gl(n,) and the associated
Lie algebra isomorphism of End(V) with gl(n,t) determined by
associating with each linear transformation on V its matrix relative to
this basis. It is easily seen that An(V ) is mapped onto the orthogonal
group O(n)  G/(n,) and that bn is mapped onto the set o(n) of
skew-symmetric matrices in gl(n,t). This provides another proof of
the fact that the orthogonal group O(n) is a closed Lie subgroup
of Gl(n,) with Lie algebra o(n).
A Lie "algebra g has a bilinear operation--the bracket [ , ]. The
automorphisms of g which preserve the bracket are precisely the Lie
algebra isomorphisms of . We shall denote them by A(O). According
to 3.54, A(O) is a closed Lie subgroup of Aut(O) with Lie algebra the
derivations of [ , ] which we shall denote by b(O).
Suppose that G is a simply connected Lie group with Lie algebra g.
Let A(G) be the group of all Lie group automorphisms of G. The map
(1) ,p: A(G)-, A(O) defined by ,p(x) - dx
is a homomorphism and is 1 : 1 by 3.16 and onto by 3.27. Thus we can
induce via ,p a manifold structure on A(G), making A(G) into a Lie
group with Lie algebra isomorphic with b(O).

II II
Lie Groups
HOMOGEITEOUS MITII=OLDS
3.8 Theorem Lt H be a closed subgroup efa Lie group G, and let
G/H be the set {oH: o e G} of left coset$ modulo H. l.et r: G --* G/H
the eatrul projection (o) -. oH. 7"hen G/H has u unigjue manifold structure
such that
(b) There exist local mooth sections of G/H in G; that is, if oH
there is u neighborhood W of oH and u C  mop  : W-- G such that
ro- 1 id.
eetooe Existence We topoloize G/H by requiring U in GJH to be
open it" and only it" r-l(U) is an open set in G. With this topology,
r is an open map since it" W is open in G, then
,,-'((W))- U wh,
which implies that r(W) is open in G/H. Moreover, GJH is HausdortT.
To see this, first observe that the set a  G x G consisting ot" all
pairs (o,) t'or which there exists an h e H such that o -- h is a closed
set since J- oc-l(H) where oc is the continuous map (o,-)
ot" G x G into G. ]Tow, it" oH and -H are distinct points ot"
(o,-) does not belong to J, o there exist open neighborhoods V ot"
and W ot" - in G such that (V × W)  A ,= . Then r(V) and r(W)
are disjoint open neighborhoods at'oH and -H respectively, which proves
that G/H Js Hausdorff. A countable basis t'or the topology on
projects under r to a countable basis t'or the topology on G/H. Thus
G/H is second countable.
Accordin to 3.42, the closed subgroup H is a Lie subgroup
Suppose that G is of dimension d and that H is ot" dimension d --
To prove that G/His locally Euclidean and to obtain a cover ot" smoothly
overlappin coordinate systems on G/H, we first prove that there exists
a cubic centered coordinate system (U,?) about • in G, with coordinate
t'unctions x .... , x,, such that distinct slices of the t'orm
(l) x ,= constant for all i e (l, ....
.lie bn distinct left cosets ot" H. Let  be the distribution on G deter-
mined by the Lie algebra ot" H. Then, by 1.60, there is a cubic centered
coordinate system (V,?) about • in G, with coordinate t'unctions
x .... , x,, such that the integral manit'olds ot"  in V re slices ot" the
t'orm (l). Since H is a closed subgroup ot" G, and theret'ore as a Lie
subgroup has the relative topology, V can be chosen small enough so
that
(2) • V  H -- the slice So through e.

Homogeneous MaMfold$ 121
Choose neighborhoods U and V 1 of e, cubic relative to the coordinate
system (V, qO, such that
(3) VIV  V and U-1U  V.
Now suppose that  and ," are points of U which lie in the tame coset
modulo H, so  6 ,-H. Then
(4)
So o  *'(Va t So). Now *'(V a t So) is an integral manifold of- which
lies in V by the choice of V, in (3), and ,-(V,  So) is connected. There-
fore ,-(VI  So) lies in a single slice of V. So  and ,- lie in the same
slice. Conversely, it is easily seen that a single slice lies on a single
coset. Thus (U,?) is the desired coordinate system.
.. t ,
Let S be the slice of 9(U) on which xt-i ..... .vd vanish. Let -1 be
the map defined by setting
¢5) -' -* , o 9-' [ s: s--, ,(v).
Then -t is one-to-one by the choice of the coordinate system (U,9),
and is also continuous and an open map; hence it is a homeomorphism.
Let  be the inverse,
(6) : ,(u) --- s  ".
Then (,r(U),t) is a coordinate system about the identity coset in G/H.
We obtain coordinate systems about other points of G/H by left trans-
lations. Indeed, if,  G, we let 'o be the homeomorphism of G/Hinduced
by left translation Io on G; that is,
(7) .(,n) ,=

122 Z.ieGroups
Then for each oH ¢ G/H we define a map re.r by setting
Then([.(r([f)), re.r) is a coordinate system about oH. Note that in
this notation rear is just the map re. Now, we claim that one obtains a
differentiable structure on G/Hby maximizing the collection of coordinate
systems
(9) re.,,):
One has only to check differentiabil/ty on overlaps. So let
(i.,(.(), re.,,,) a.e (.,(.(), re... ) be o s.¢h coordinate ,.s,
and let
el0)
We must prove that re, o rerl V is C ".  t e V. Then since
-.-' ° .s ° re -s(t)  r(U), there exists an element g  H such that
-sos?-s(Og  U. It follows that there exists a neighborhood W of"
t in V such that -ss?-s(W) U. It suffices to prove that
re, o re;r I W is C ®. But we can expr'ws re,,u o re|su |W as the
following composition of" C ® maps: .
where r o is the canonical projection of ?(U) onto S. Thus
re.,,, o I v is c
With this differentiable structure on G/H, the projection r:
is ¢®; indeed, r restricted to an open set of the form l.(b') is nothing
other than the composition re;r o r o o ? o !.-, I I.(U). Thus result
(a) holds; and as for (b), I. o -o re.r is a local smooth section of
GIH in G on the neighborhood l,(,r(Cr)) of H.
Uniqueness Let (G/H denote G/H with another differentiable
structure satisfying (a) and (b).
Then the identity map and its inverse are both C ® since locally they can
be expressed as the composition of local smooth sections into G
followed by .. Thus id is a diffeomorphism, which proves uniqueness.
3.59 Definition Manifolds of the form G/H where G is a Lie group,
H is a dosed subgroup of G, and the manifold structure is the unique one
satisfying (a) and (b) of Theorem 3.58 are called homogeneous manOeold.

llomogeneous anifolds 123
$.60 Remark Observe that f is a C  function on G/H if and only if
fo r is a C  function on G; for iffis C , then certainlyfo r is C . Con-
versely iffo r is C ®, then f, which can locally be represented as the composi-
tion of a smooth section ofG/Hin G withfo r, is also C ®.
3.61 Definitiom Let
(1) t/: 6 X M-* M
be an action of 6 on M on the left (cf. 3.44); and, as usual, let
(2) n.(m)-
The action is called effective if e is the only element of G for which , is
the identity map on M. The action is called transitive if whenever m and n
belong to M there exists a a in G such that ,(m) -- n. Let me . M, and let
(3) H l G: - me}.
H is a closed subgroup of G called the isotropy group at me. The action /
restricted to H gives an action of H on M on the left with a fixed point me,
so by 3.45 we have a representation
(4) : H--* Aut(M,) where () -- d/, [ M,.
The group (H) of finear transformations of M, is called the linear isotropy
group at me.
$.62 Theorem Let t l: G x M-- M be a tranMtive action of the Lie
group G on the manifold M on the left. Let me  M, and let H be the isotropy
group at me. Define a mapping
(1) t:GIH--M by
Then  i a diffeomorphism.
gooF Observe first that
t/,(me) for any h  H.  is surjectiv¢ since G acts transitively, and is
injectiv¢ since if (H) -- (H), then ,-l,(me) -- me, which implies
that - belongs to H, or in other words that aH- H. In view of
Exercise 6 of Chapter l, it suffices to prove that  is C  and that d
is non-singular at each point.
According to Remark 3.60,  is C ® if and only if  o r is C ® where
r is the natural projection of G onto G/H. But
(2)
where ira, is the C ® map of G into G x M defined by
(3) i,.() -- (,m0).
Thus  is C ®.

124 £/e Groups
Now kt --J o . Sinea the kernel of drlG . is (.. = G., in
o to  at   (/.n is n-n it is scnt to
t  kei of dO  G. b  (,.. S for
<4) o
it su to pve t the kernel of dG. is
kd  G.   x e ked  G. To
only p at p X  H for   e  whe X is the  invt
vtor field on G dsten by x. For ts, it is sut to
at c t or to c c t  (cxp
, or cn (cxp  m , wch pli at p
.  nscnt vor to  c at  is
TErn  is evehe o!, and e
3.63 lemm'ka If G is a Lie Stoup and H a dosed subgroup of G,
then there is a natural action 7ofG on the homogeneous manifold G/Hon the
left, namely,
(z) T: (; x (;IH--,. IH, -:-
It is easily checked that 7 is C ® and indeed gives an action of G on G/H on
the left. Moreover, it is obvious that 'is a transitive action. Now, for each
o in G, 7, (notation as in 3.61(2)) is a diffeomorphism of G/H; and given any
two points .H.ind ),H of G/H there is a diffeomorphism ,_ taking ),H to
.H. The reason that manifolds of the form G/H are called homogeneous is
that they possess this transitive group of diffeomorphisms. Conversely,
Theorem 3.62 shows that if a manifold M has a transitive group of diffeo-
morphisms in the sense of 3.61(I), then M is diffeomorphic with a homo-
8eneous numifold G/H.
From Theorem 3.62 we get another characterization of the manifold
structure on G/H, namely, tho set G/H ha a unique manifold structure such
that tho natural map (I)
3.64 Theorem Let G be a Lie group and H a closed normal subgroup of
G. g'ben tho homogeneoua manifold G/H with its natural group structure Is a
Lie group.
ZtOOF One has only to check that the map
(I) (oH,H) 
ofGiHx GJH--GJHis C °. Let %: W.-G and a: Wv-*G he
local section of Gill in G on neighborhoods W. of oH and Wv of 'H

llomogeneous Manifolds 12
respectively. Then locally the map (1) can be expressed as the following
composition of C ® maps:
(2) ,r o q o (% x ,),
where ?: G x G - G is the map ?(,) --
$.65
(a)
Examples of homogeneous manifolds
Let {e,: i == 1 .... ,n} be the canonical basis of IR" where e, is the
n-tuple consisting of all zeros except for a 1 in the/th spot. Each matrix
a EGI(n,.) uniquely determines a linear transformation on
which we shall also denote by a, by requiring that
(1) (e) --  .
In other words, if we consider the n-tuples of IR" as n x 1 matrices,
then a acts on IR" in the natural way by matrix multiplication. The
map (a,v) -. a(v) gives an action of Gl(n,.) on II" on the left:
(2) 6(n,) x "  '.
Let ( , ) denote the standard inner product on II" with respect to
which the basis {e} is orthonormal. Then if a E Gl(n,[),
(3) (v).w) -
If a  O(n), then ata .. I, so it follows from (3) that
(4) (v),(v)) - v,(v)) -
thus  preserves lengths of vectors. Thus the action (2) restricted to
O(n) x S "-a factors through the unit sphere S -a.
(5)
According to 1.32, the factoring map is smooth, so we have a natural
C ® action
(6) O(n) x
of the orthogonal group O(n) on the unit sphere $,-x on the left.
We claim that the action (6) is transitive. Ifvx ¢ S "-x, let {vx, vs,..., v=}
be an orthonormal basis of ." containing vx as the first element. Let
(7) , -  ,.

126 Lie Group
(b)
Then the matrix o whose entries are determined by (7) is orthogonal,
and
(8) o(e,) -- v,.
It follows that if v and w are any two points of $'-,  is an orho 8-
onal matrix  such hat (v) -- w. Thus the action (6) is transitive.
The set of matrices in O(n) of the form
(9)
forms a closed subgroup of O(n). The matrice 0 occurring in this
subgroup are precisely the matrices in O(n -- 1). So O(n -- 1) sits
in O(n) as this natural closed subgroup. We claim that O(n -- 1) is
precisely the isotropy group for the action (6) at e, c $.-t. Clearly
the elements of O(n -- 1) leave e fixed. On the other hand, suppose
that  c O(n) and that o(e,) R e,. Now
o(e,) -" .
so o. R 0 for l  n and o...- 1. Since o is orthogonal, oo  -- 1,
and this implies that  o'. -- 1. Since on. -- 1, we must have 0.4 -- 0,
i < n. Thus o 60(n -- 1). It follows from Theorem 3.62 that the map
(10) O(n)/O(n -- 1) - $'-', oO(n -- 1) -.
is a diffeomorphism. Tms the aphere $,_t is in a natural way dlffeo-
morphic with the homogeneous manifold O(n)/O(n -- 1).
By a similar arsument one can show that there is a diffeomorphism
of the aphere S "- with the omogeneou manifold $O(n)/$O(n -- 1).
Let (e: l . 1 .... , n) be the canonical complex basis of C" where
e, is the n-tuple consisting of all zeros except for a 1 in the ith spot.
Just as in Example (a), each matrix o c GI(n,C) uniquely determines a
linear transformation of C', which we still denote by o, by requiring
So if we consider the complex n-tuples of C" as n x 1 matrices, then o
acts by matrix multiplication. The map (o,v)-. o(v) is an action of

Homogeneous Manifolds 127
the real 2n*-dimensional Lie group Gl(n,C) on the real 2n-dimensional
manifold C" on the left:
GI(n,C) x C '= -- C*.
, ) denote the standard inner product on C', where
(12)
Let<
(13)
Then if o E GI(n,C),
(14)
<o(v),w) --
If  is an element of the unitary group U(n), then to == I, so it follows
from (14) that  preserves lengths of vectors in C'. If X denotes the
unit sphere in C', then the action 02) restricted to U(n) x X factors
through X, and so, by 1.32, yields a C = action
05) U(n) x X--,. X
of the unitary group U(n) on the unit sphere X ¢ C" on the left. It
follows by an argument similar to that in Example (a) that the action
(15) is transitive and that the isotropy group for the action (15) at the
point e. is simply U(n -- 1), which we consider to be a closed subgroup
of U(n) by identifying #  U(n -- I) with
(16)
0
0 "" 1
Thus, by 3.62, X is difl'eomorphic with the homogeneous
in U(n).
manifold U(n)/U(n -- 1). But now under the canonical global coordinate
system on C" (fn by the dual basis to the real basis {ex ..... e.,
/-e ..... f-e.} of C'), X is diffcomorphic with S ="-x = =".
Thus the sphere S ="- is diffeomorphic with the homogeneous manifold
u(n)/u(n- I).
By a similar argument, the sphere S "- is also diffeomorphic with the
homogeneous manifold SU(n)/SU(n- 1). In particular, since SU(I)
consists only of the 1 x 1 identity matrix, S  is diffeomorphic with
SU(2). Thus S  can be given a Lie group structure. It can be shown
that S  and S 3 are the only spheres possessing Lie group structures [25].

(c)
(d)
The realprojectlve space p,-s is the set of equivalence cla of points
in R" -- {0} whore (a s .... , a=) is equivalent to (bl, .... b) if there
is a non-zero  number c such that ca -- b for i -- 1,..., n. If
p,-z is given the largest topology in which the natural projection
07) ,,: (R- -- {0}) -,.."--1
is continuous, then ,r restricted to S-: is a 2-fold covering of P'-Z,
and it is easily established that P'-a  a unique differentiable structure
such that this covering map is locally a difl'eomorphism. By an argument
dmilar to that in Example (a), one can show that P-a is difl'eomorphic
with the homogeneous space SO(n)/O(n- 1), where we condder
O(n -- 1) as a closed subgroup of SO(n) by identifying O e O(n -- 1)
with the following element o of SO(n):
A a t, the complex projective space CP "-1 is the set of equivalence
elapses of points of C" -- {0} under the equivalence relation in which
(a .... , a.) is equlvalcnt to (/ .... ,/.) if thor is a non-zero
complex number 7 such that 7a, --/, for i .- 1,..., n. We make
CP'- into a real 2(n- 1) dimensional manifold as follows. Tho
action of the Recial unitary group on the unit sphere in C" preserves
these equivaknce classes, and each element of CP -t has representatives
of unit length; thus we have a natural transitive action of SU(n) on
the set CP-:
(18) SU(n) x CP"- CP'-z.
The subgroup of SU(n) leaving fixed the point of CP "- determined by
c,  C" is simply U(n -- 1), which we consider to be a dosed subgroup
of gU(n) by identifying # 6 U(n -- 1) with
0 "- 0 l/(det#)
It follows that the map
oU(n -- I) P* {o'(e.)},

Homogeneous Manolds
where {¢(e)} denotes the point of CP *-x determined by ¢(e), is a well-
defined one-to-one map of SU(n)/U(n- 1) onto CP 'x. We give
CP -x the structure of a real 2(n- 1) dimensional manifold by re-
quiring that the map (19) be a diffeomorphism.
Let V be a real d-dimensional vector space, and let $,(V) be the set
of p-frames in V. That is,
(20) S,(V) -- {' -- (wl ..... w): the wt .... , w,
are linearly independent elements of V}.
If we choose a basis v x ..... v for V, then Gl(d,[R) acts on V by matrix
multiplication. We define
: G/(d,r) x S,(V) S,(V)
(20
by (,,, ..... = . . . ,
Observe that if , •  S,(V), then there is a   Gi(d,,) such that
t/(,) == . Now let" be the element of $=(V) determined by the firstp
elements of the basis v t ..... v=; hence  == (v=,..., v=). Let H
be the subset of GI(d, IR) leaving  fixed. Then
1 A
where I is the p x p identity matrix; thus H is a closed subgroup of
GI(d,R). It follows that the map aH-* t/(,') is a one-to-one map of
the homogeneous manifold GI(d,R)/H onto the set $(V). We give
$(V) the structure of a d" p-dimensional manifold by requiring that this
map be a diffeomorphism. It is easily checked that this manifold
structure is independent of the basis of V chosen. $(V) is called
the Stiefei manifold of p-frames in V.
Again let V be a real d-dimensional vector space, and now let Mr(V)
be the set of all k-dimensional .subspaces (k-planes) of V. If we choose
a basis v=,..., v= for V, then the orthogonal group O(d) acts naturally
on V by matrix multiplication; and since non-singular linear transforma-
tions map k-planes to k-planes, we have a map
(23) t/: O(d) x Mr(V)--* Mt(V).
Observe that ifP and Q are k-planes, then there is a a E O(d) such that
t/(,P) == Q. Now, let Pe be the k-plane spanned by the first k elements
of the basis v= ..... v=, and let H be the subset of O(d) leaving Pe fixed.
Then
(24) H -- { (-)  O(d):   O(k), , O(d-- k)},

130 Lie Groups
so H is a closed subgroup of O(d) which we can identify with O(k) x
O(d - k). Then the map o(O(/c) x O(d --/c))  No, P,) is a one-to-one
map of the homogeneous manifold O(d)/[O(k),x O(d--k)] onto
the set M,(V). We make Mr(V) into a (d -- k)k dimensional manifold
by requiring that this map he a diffeomorphism. One can check
that this manifold structure on M,(V)is independent of the basis
chosen for Y. M,(V) is known as the Granonn manifold ofk-planes
in Y.
$.66 Propmitton Let H be a closed subgroup of the Lie group G. If
H and G/H are connected, then G Is connecled.
etoov Assume that
where U and V are non-empty open subsets of G. Then
(2) G/H = Hu) 
where v(U)and V)are non-empty open subsets of G/H, Since
is connected, there must he a point oH of G/H such that
(3) oH Hb3 rt H.
Now, (I) implies that
(4) oH =, (oH r b')  (oH r
where (oH r b') and (oH r V) are both open subsets of oH (since
H has the relative topology). According to (3), both (oH r b') and
(oH r V) are non-empty; thus, since oH is homeomorphic with H
and therefore connected, we have
(5) (oH r b3 r (oH r V)  e,
which implies that
(6) u v ,
which proves that G is connected.
$.67 Theorem Each of the Lie groups SO(n), SU(n), and U(n) s
connected for n  I, and O(n) has two components (n  I).
Pltoo SO(I) and SU(I) are connected since they both consist only
of the ! × ! identity matrix, and U(1) is connected since
() - ((A): A  c, IAI - 1}.
That SO(n), SU(n), and U(n) are connected for all n now follows from
3.66 by using induction on n and the representation of spheres as homo-
geneous manifolds siren in 3.6&

Homogeneous Manifolds 131
Since every matrix in O(n) has determinant 4-1, the orthogonal group
can be written as the following union of two non-empty disjoint con-
nected open subsets:
O(n) - SO(n) u SO(n)
Thus O(n) has two components.
where
Theorem 61(n,) has two components.
eoo Let 61(n, )+ be the subset of 61(n, ) consisting of matrices
with positive determinant, and let 61(n,)- be the subset consisting
of matrices with negative determinant. Gl(n,) + and Gl(n,)- are
disjoint homeomorphic open subsets of 61(n,), so it suffices to prove
that Gl(n,) + is connected. To do this, we shall show that each element
of 61(n,)+ can be joined to the identity matrix by a continuous curve.
First we show that each element of 61(n,.) has apolar decomposition;
that is, each matrix  e 61(n,) can be expressed in the form
(1)  -- PR
where P is a positive definite symmetric matrix and R e O(n). (Recall
that all the eigcnvalucs of a symmetric matrix are real, and that a sym-
metric matrix is positive definite if each of its eigcnvalucs is strictly
positive.) Since (o) ' -- o , o  is symmetric. Let a be an eigenvalu¢
of o  with eigenvector v  II'. Then, according to 3.65(3), if ( , )
denotes the standard inner product in
a(v,v) - ((v),v)- (v),(v)).
Consequently, a : O, and since  is non-singular, we must have a > O.
Thus  is a positive definite symmetric matrix. Since  is symmetric,
there exists an orthogonal matrix   O(n) such that the matrix
is diagonal (cf'. Exercise 22(a)). Since the eigcnvalues of" t are all
positive, the matrix (2) has a square root
tlut is, (3) is a diagonal matrix, and cach diagonal entr is the positive
square root of the corresponding entr in the matrix (2). Let
(4) ' -

P is a positive definite symmetric matrix, and R is orthoonal since (4)
implies that
(6)
" P-ooP -  P-PPP- " I.
Thus o -- PR, a we a.rted in (I).
If o  Ol(n,) +, then e has a polar decompos/tion (1), where now
R must have positive determinant; thus R 6 SO(n). Let
(7) .= tl + (t -
for t 6 [0,1]. Then Pt is positive definite for each t, so the path t ,- PtR
is a continuous curve in Gl(n,) ÷ joining o to R. Since SO(n)is con-
nected, and therefore pathwise connected, R can be joined to the identity
matrix I by a continuous curve. Thus Gl(n,) + is pathwise connected,
which completes the proof that Gl(n,r) has two components.
EXERCISF
Prove that a connect Lie group is automatically second countable;
that is, the assumption of second countability in the definition of
connected Lie group is redundant.
Show that the examples in 3.3 are Lie groups.
Prove that the examples in 3.5 are Lie algebras.
Supply a proof for Proposition 3.12.
Prove the necessity of the connectedness aaumption in 3.16.
Let .P be an ideal of forms on G* generated by a collection
(o,..., o,.4) of independent left invariant l.forms. Let
be the d-dimensional subspace of the Lie algebra of G annihilated
by the o (l .= 1,..., c -- d). Prove that l) is a subalgebra of I if
and only if.P is a differential ideal.
In this exercise we outline the proof of Theorem 3.23.
(a) Let : (,Y, xo)--(Y, yd be a covering, and let 0t: (Z,zo)--
(¥,y,) be a continuous map where Z is pathwise and locally
pathwise connected, and where a, Or:(Z,zo) )  r,(rt(y, x0)).
Prove that there is a unique continuous map : (Zz0)--
(,Y, x0) such that r o

8
(b)
F.rcises 133
(c)
(Sketch: First prove part (a) for the case in which Z is the
unit rectangle [0,I] x [0,I] and zo -- (0,0). Here the condition
on the fundamental gr6ups is trivially satisfied, and the proof
follows by partitioning the unit rectangle fine enough so that
each subrectanglc of the partition maps under  into an evenly
covered set in- Y. As a particular application of this case, it
follows that each path in Y has a unique lift to X once base
points are chosen.
For the general case, define the lifting  of e as follows.
Let z  Z, and let y: [0,1] --* Z be a path joining :e to :; thus
7'(0) = ze and 7,(I) = z. Then ec o 7' has a unique Eft  to X
such that (0) = xe. Set (z) = (I). Now use the condition
on the fundamental groups and the unique lifting property
already established for rectangles to show that & is well-
defined, independent of the choice of the path 7' from :e to :.
Finally use the local path-connected property of Z to show that
& is continuous.)
If X is a pathwise connected, locally pathwise connected, and
semi-locally I-connected topological space, then X has a
simply connected covering space.
(Sketch: Fix a base point xe G X. We define an equivalence
relation on the set of all paths in Xwith domain [0,1] and with
initial point xe. Two such paths Yi, ya will be called equivalent
if ,l(l) =: y,(I), and the loop yt-ly at xe obtained by first
traversing y and then traversing y, in the reverse direction is
homotopically trivial at xe. The point set of the simply con-
nected covering space A ° of X is the set of such equivalence
classes {y}. The projection map ': AO"X is defined by
-({y})  y(1). Let {y} G A °, and let Ube an open neighborhood
of y(l) in X. Let U({y}) consist of all elements {-} of A °
possessing a representative path - which first traverses y and
then remains in U. Prove that the collection of such sets U({y})
forms a basis for a topology on A ° in which A ° is simply con-
nected, and in which r: AO--* X becomes a covering.)
A covering of a simply connected space is a homeomorphism.
(This is an immediate application of part (a).)
Prove that if -: J0 --* M is a covering of a differentiable manifold
M, then J0 is second countable. (It suffices to prove that rt(M,m)
is countable. This follows easily from the observation that M
can be covered by a countable number of open sets, each homeo-
morphic with an open ball in Euclidean space.)

9 Let O and H be Lie groups with 0 connected. Prove that a C
map q: 0 -- h r senclin the identity of G to the identity of H/s a
homomorph/mn/f ,)? pulis left inv.,t ,forms on H back to
/nvariant forms on 0. Show that the assumption of counectedness
for 0 is necessary.. (Hint: Osc can construct a differential ideal
on O x H as in 3.15($) whose maximal connected integral sub-
manifold ! through (e,e) is a Lie Subgroup .of O x H. Factor the
raph of ? through I, and show that the natural covering homo-
morphism of I onto O is ! : !.)
10 Prove that is not e a for any A 
11 Let o(t) and p(t) be smooth curves in a Lie group O such that
o(0) -- p(0) -- e, and let c(t) -- o(t)#(t). Prove that
- +
(Hint: Consider the group multiplication map : O < O -- G, that
is, (y,-) -- y-. Let v, w  O,, and show that
d,(o,.,) - + (0,-)) -- o +
12 Let G be a connected lie roup, and let ?: G- H be a homo-
morphism with a discretekernel. Prove that the kernel lies in the
center of O. Use this fact to prove that the fundamental roup of a
lie group is abelian.
13 Prove that Exampie 3.$(d) is, up to isomorphism the only 2-
d/mensional non-abelian Lie algebra. Conclude that Example
3.3(h) is, up to isomorphism, the unique simply connected 2-
dimensional non-abehan lie group.
14 Show that there are matrices A, B  gl(n,C) such that
15 Prove that the exponential map for GI(n,C'.) is surjective. (Outline:
First, by using the Jordan canonical form, reduce the problem to that
of showing that each elementary Jordan matrix (a matrix with a fixed
constant  on the diagonal, with l's immediately above each
diagonal entry, and with zeros elsewhere) in GI(jo.), for ! j
is the exponential of an element of gl(j,). Let A be an elementary
Jordan matrix. Write A as (L/) • N where the diagonal entries of
are 1's. Prove that  -- e  and that .,V -- • ., where AAt --
To find At, observe that since sufficiently highpowers of the difference
I -- N vanish, N has.a lc,arithm, namely,
logN-- -- (/-N).
Use a formal power series argument to show that • s N = N.)

F.ercises 135
16 Let G be a Lie group. A vector field Y on G is right invariant if
is re-related to itself for each ¢  G. Prove that the set of right
invariant vector fields on G forms a Lie algebra under the Lie
bracket operation and is naturally isomorphic as a vector space with
G,. Let ?: G-*.G be the diffeomorphism defined by ?() = o -1.
Prove that if X is a left invariant vector field on O, then dq(X) is
the right invariant vector field whose value at • is --X(e). Prove
that X t- dq(X) gives a Lie algebra isomorphism of the Lie algebra
of left invariant vector fields on G with the Lie algebra of right
invariant vector fields on O.
17 Find an example of a Lie group G with a Lie subgroup A that is not
closed in O.
18 Let O be an n-dimensional abelian connected Lie group. Prove there
is an integer k, 0  k  n, such that O is isomorphicwith -t
where T t is a torus of dimension k. First, prove that up to iso-
morphism,  is the unique simply connected abelian Lie group.
Next, show that if D is a discrete subgroup of A n, then either D
{0} or there is an integer k, 1  k  n, and k linearly independent
vectors vl, • • •, vt in  which generate D. To prove this, suppose
that D # {0}, and let k be the smallest integer such that D is con-
tained in a k-di.mensional subspace Vof . Find a basis wl .....
for V such that the w  D. Let
A, -- rw: 0  r  1, r, > 0
for/-- 1 ..... k. Choose v to be an element ofA  D with smallest
possible coefficient r. Then {t, 1 ..... t,t} will be a linearly inde-
pendent set generating D.
19 Supply a proof for 3.56.
20 Prove that the group of automorphisms of a connected Lie group
G is itself a Lie group. (Outline: Let ¢ be the simply connected
covering group of G, and let - be the covering homomorphism.
Let D  ker -. We know from 3.57(b) that the group A(¢) of
automorphisms of ¢ is a Lie group. Show that the group of auto-
morphisms of (7 is naturally isomorphic with the (closed) subgroup
of A(U") consisting of those automorphisms of ¢ which map D
onto D.)
21 Prove that U(n) is diffeomorphic with S  x SU(n).
22 (a) Deduce from a "super-triangularization" argument similar
to that following 3.35(6), but now using unit eigenvectors and
orthogonal complements, that if A is a hermitian matrix
(that is, A is complex and AT .- At), then there exists a unitary

136 L Gvoup
matrix  such that A "I is d/aonal. Bb' a s/milar argument,
prove that/f A is a real symmetric matrix (A -- A'), then there
exists a real orthosonal matrix  such that A -s is diagonal.
(b) Using part (a) and 3.46(7), prove that the oxponent/al map
maps the hermitian matrices one-to-one onto the set of pos/tive
deiln/te (all genvalnes positive) hermJtian matrices. Deduce
also that the exponential map maps the real symn-tric matrices
one-to.one onto the set of positive deilnJte symmetric matrices.
23 Use part (b) of Exercise 22 to prove that the polar decomposition
3.68(I) is unique. (HIn: If u = P.q =, PR, iirst prove that
P = PI s, then apply part (b) of Exercise 22 to P and PI .)
24 Prove that every matrix u  (71(n,C) can un/quely be written as a
product u  PR, where P is a positive deilnite hermitian matrix
and R is a un/tary matrix.
:73 Prove that 71(n,C) is connected.
26 Complete the details of $.65(b) and (©).

We shall consider integration of p-forms over differentiable singular p-chains
in n-dimensional manifolds, and integration of n-forms over regular domains
in oriented n-dimensional manifolds. For both of these situations we shall
prove a version of Stokes' theorem. This is a generalization of the Funda-
mental Theorem of Calculus and is undoubtedly the single most important
theorem in the subject. We shall also consider integration on Riemannian
manifolds and on Lie groups. Finally, we shall introduce the de Rham
cohomology groups and shall prove the Poincar lemma, from which we
will conclude that the de Rham cohomology groups of Euclidean space are
trivial. This lemma will be of central importance for the de Rham theorem,
which is stated at the end of this chapter and proved in Chapter 5.
ORIENTATION
4.1 Definitions Let V be a real vector space of dimension  n. The
notion of an orientation on V was introduced in Exercise 13 of Chapter 2.
Recall that the nth exterior power An(V) is I-dimensional, so that An(V) -- {0}
has two components. An orientation on V is a choice of a component of
An(V) - {o}.
Now let M be a connected differentiable manifold of dimension n. We
shall call M orientable if it is possible to choose in a consistent way an
orientation on M.,* for each m  M. More precisely, let 0 be the "O-section"
of the exterior n-bundle A.*(M); that is,
(I) 0 = 0 {OAn(M*m)}.
Then since each An(M*)- {0} has exactly two components, it follows
easily that A,*(M)- O has at most two components. We say that M is
orientable if A,*(M) -- O has two components; and if M is orientable, an
orientation on M is a choice of one of the two components of A*,(M) -- O.
138

Orientation 139
A non-connected manifold M is said to be orientable if each component of M
is orientablc, and an orientation is a choice of orientation on each component.
Let M be oriented, and let v I ..... v. be a basis of M. with dual basis
s,..-, .. We say that the (ordered) basis v s ..... v. is oriented if
s ^ """ ^ . belongs to the orientation.
Let M and N be orientable n-dimensional manifolds, and let 10: M--* N
be a differentiable map. We say that 10 preserves orientations if the induced
map 1o: A*(N)--* A*(M) maps the component of A*(N) -- 0 determining
the orientation on N into the component of A*(M) -- 0 determining the
orientation on M. Equivalently, 10 is orientation-preserving if dlo sends
oriented bases of the tangent spaces to M into oriented bases of the tangent
spaces to N.
4.2 Proposition Let M be a
Then the following are equivalent:
(a)
dierentiable manifold of dimension n.
M i$ orientable.
(b) There i$ a collection 0 = {(V,10)} of coordinate system on M uch
that
(1) M-- U V and det_-x >0 on UV
hener (U, x, ....
(c) ere i a nowhere-ishing n-form on M.
 We n sume, without loss of gity, thnt M is conn.
We prove that (n)  (b)  (c)  (a). Giv (n), at M i otable,
choose nn orientation on M; nt i, we choo one of e two m-
nents, ! it A, of A(M)--O. Obse tbat for ch mM,
A  A(M) is prissy one of e two mpont of A(M) -- {0}.
Now ! • consist of ! of those coordinn sytem (V, y ..... y)
on M uch thnt the mnp of V into A(M) dn by
(2) m  (dy A . . . A @)(m)
has range in A. Now, if (U,
two coordinnte systems on M, then for m  U  V,
Oy, (dy ..-  dy,)().
If th coornate systs long to , th nessadly
d[
>°
for ench m  U  V. Consuently, (I) is satisfied, d sult (b) follows
from (n).

I 0 Ine'alon on Manlf old
Now assume (b). Let (?,) be a partition of unity subordinate to the
cover of M given by the coordinate neighborhoods in the collection
• with ?, subordinate to (V,, x1',. • •, x.r). Then
is a global n-form on M, where ?, dx1' A "- A dx.' is defined to be the
0-form outside of F,. That ¢0 vanishes nowhere follows from the fact
that for each m, re(m) is a finite sum with positive coefficients of elements
of one component of A,,(M*) -- {0). Thus (c) follows from (b).
Finally, let ¢0 be a nowhere-vanishing n-form on M, and let
+ - U (a): a  , a > o),
A- = U {am): a  , a < 0}.
Then A*.(M) -- O is the disjoint union of the two open subsets A +
and A-, so A*.(M) -- O is disconnected, and M is orientable.
4.3 Exmnples
(a) Every lie group G is orientable, for if % .... , ¢0. is a basis for the
left invar/ant l-forms on G, then % A ". A ¢0. is a global nowhere-
vanishing n-form on O.
(b) The standard orientation on the Eucfidean space ' is the one deter-
mined by the d-form dr A... A
(C) Let X be a d-dimensional manifold, and suppose that there exists an
immersion f: X--* IR +. A normal vectorfleld along (X,f) is a smooth
map N: X--* T(IRd+) such that for each p ¢ X, the vector N(p) lies
in (+)t, and is orthogonal to the subspace df(X,)
Such a manifold X is orientable if and only if there is a smooth
nowhere-vanishing normal vector field along (X,f). (See Exercise I.)
(d) As an immediate appfication of Example (c), the sphere S  is orientable
for each n  I.
(e) The real projective space P" is orientable if and only if n is odd. (See
• Exercise 2.)
INTEGRATION ON MANIFOLDS
4.4 Integration in the Euclidean Space " We assume that the reader
is familiar with some theory of integration in IR'. Since we shall be inte-
grating continuous (in fact usually C ®) functions over nice subsets of "

Integration on Marigolds 141
(polyhedra for example), the theory of the R/emann ing! will be quite
sufficient. The principal theorem that we need to recall is the change of
variables formula. Several versions of this formula and their proofs may be
found in [6], [18], or [29]. A version sufficient for our purposes is the follow-
ing. Let q be a diffeomorphism of a bounded open et D in IR  with a
bounded open -t q(D). Let Jq denote the determinant of the hcobian
matrix of q:
Letfbe a bounded continuous function on q(D), and let A be a nic subset
of D. (A will be polyhedral in most of our applications. Cnerally, for the
R/emann theory, a nic subset would be one which has Jordan content.)
Then
4.5 Integration ofn-forms in IR  Asusual, weletri ..... r denote the
canonical coordinate system on IR . The Standard orientation is deter-
mined by" the n-form dri ^-.. ^ dry. Now let ¢o be an n-form on an open
set D  I n. Then there is a uniquely determined functionfon D such that
¢o ffi f dr ^ • .. ^ dry. Let A  D. We define
(1) f:
provided that the latter exists. We can restate the change of variables formula
4.4(!) in terms of differential forms. Let q, D, and A be as in 4.4, and let
o be an n-form on q(D). Then
(2) f,.,o ffi 4-f.q(¢o),
where one uses "+" if q is orientation preserving and "--" if q is orientation
reversing.
4.6 Integration over Chains The first type of integration that we
shall consider on general manifolds involves integration of p-forms over
differentiable singular p-chains in an n-manifold.
For each p  I we let
{ . }
(1) A" ffi (a ..... a,)  IR':  a,  1, and each a, : 0 .
A, is called the standardp-simplex in I,. Forp ffi 0, we set A e equal to the
l-point space {0}; A e is the standard O-simplex. Let M be a manifold. A
differentiable singularp-simplex  in M is a map ¢ of A, into M which extends
to be a differentiable (C ®) map of a neighborhood of A, in IR, into M.

142 Inte,atlon on Manlfold
In this chapter we shah refer to such a ¢ as simply a F-simplex ¢ in M. (In
Chapter $ we shall deal, in addition, with continuous singular simpfices,
and so there we shall need to retain the term "different/able" for distinction.)
A O-simplex in M cons/sts of a map of the l-point space (0) into M. AF-
c in M (with real coeients) is a finite linear combination c-
ofF-simplices , in M where the a, are real numbers.
For each p : 0 we define a collection of maps k,,: A, _.,. As for
0  i  p + I as follows:
for p =, O, /'(0) =, I and k(O) ,= 0;.
(2) k(as,..., a) ,= I -- a, al, .... a and
for p , I, | k(a," ,a,)-- (al,...,a,O,a,...,a)
If o is a p-simplex in M with p  I, we define its th face, 0  i  p, o the
(J -- 1) simplex
(3) o  '=  o k -s,
and we define the boundary of o to be the (p -- I) chain
(4)  -  (-1)'o .
We extend the boundary operator linearly to chains. Note carefully the use
of the superscript to denote faces. Thus the boundary of the p-chain
k
(-1)a¶
(0,. I) *
(0, 0) I (i, 0)
We claim that
(5)
(p : o; i j).

Integratlon on Manlfold 143
For p -- O, check the three possible cases separately. For p : I, observe
that both sides of (5) give the following maps:
1  | <. (al, .... a,)t--,(a,...,a_,O,a,...,a._O,a, .... a:)
1  t --j (ax,... ,a,),--,(a,,...,a_,O,O,a,... ,a,)
(6) 0 ,<j (a,, ,a,)-(l--, .., .. )
It follows from (5), (3), and (4) that the boundary of the boundary of
a chain is always 0. That is,
(7) =0.
Now let o be ap-simplex in M, and let ¢o be ap-form defined on a neighbor-
hood of the image of o. In most of our applications we shall deal only with
smooth forms, but for the purposes of the following definition it would be
quite sufficient for ¢o to be a continuous p-form. First of all, ifp ---- 0, then
a 0-simplex consists simply of a point in M, and a 0-form is simply a function.
In this case, we define the integral of the function ¢o over the 0-simplex o to
be the value of ¢o at the point o(0) ¢ M:
(s) =
If p  1, then since o extends to be a smooth map of a neighborhood of
A, in " into M, ¢o can be pulled back via o to ap-form &(o) on a neighbor-
hood of A,. In this case, we define the integrai of the p-form ¢o over the
p-simplex o by
(9)
We extend these integrals linearly to chains, so that if c  _ ao, then
Perhaps the single most important theorem in integration theory on
manifolds is Stokes' theorem. This is a generalization of the Fundamental
Theorem of Caiculus. Observe that in our context the Fundamental Theorem
says that if F is a smooth function on the real line, and if o is a smooth
l-simplex in the real line, then
(11)  =fF.
We shall present two versions of Stokes' theorem. The first is in terms of
integration of forms over chains.

144 Integration on ManC'olds
4.7 Stokes' Theorem I let c be a p-chain (p  1) in a differentiable
manifold M, and let ¢0 be a smooth (p -- 1) form defined on a neighborhood
of the Image of c. Then
lit(x)]: It is sufficient to consider the case in which the p-chain ¢
consists of a single p-simplex u. Thus we must prove that
(2) idce --f,ce.
It follows immediately from our definitions that (2) is equivalent with
Observe first of all that the case p -- 1 reduces directly to the Funda-
mental Theorem of Calculus:
(4)
-r (co o o) dr -- o(o(1)) -- ce((O)),
so we now assume that p : 2. Then the (p -- 1) form o(o) can be
expressed as
(5) (¢o) -- , a s drx ^"" ^ r s ^". ^
where the circumflex over a term means that the term is to be omitted,
and where the a s are C ® functions on a neighborhood of
Since the integral is linear, we may consider the special case in which
o(¢o) consists of a single term of the form a s dr I ^.. • ^ dr ^... ^ dr,.
In this case, the left-hand side of (3) becomes
(6) (--1)'-I f --t dr, ^"" ^dr,.
J" irs
To evaluate the right-hand side of (3), we observe that for 1  i  p,
s (1  j  i -- 1)
(7) k-X(r,) *" (J *" 0
tr_x (t + 1 < j < p),
and
/1 --.. ,, (j- 1)
I
,rs_t (1 < j  p).

Integration on Manlfold 145
Applying (7) and (8) to the right-hand term of 0), we obtain
(-)' ,_,-'(a,
(9)
+ ¢--l)'fa ,ah,...,r,-`, 0, r .... ,
A""
A
dr_l.
We shall now apply a change of variables to the first term on the right-
hand side of (9). Let % be the diffeomorphism of IP, -1 defined by
1 - r, r ..... r.4 (i '= 2)
Then %('-) ,, -, and I,/%1 ,, 1, so that by 4.4(1), the first term
on the right-hand side of (9) is equal to
4.8 Integration on an Oriented Manifold Let M be an n-dimensional
oriented manifold. We shall integrate n-forms over regular domains in M.
A subset D of M will be called a regular domain if for each point m  M one
of the following holds:

146 Inteaffon on Manold
(a) There is an open neighborhood of m which is contained in M -- D.
(b) There is an open neighborhood of m which is contained in D.
(c) There is a centered coordinate system (U, qQ about m such that
(U  D)- ¢p(U) H , where H" is the half-space of
defined by r..  O.
Points of D of type (b) are called interior points and comprise the interior,
Int(D), of D. Points of type (c) are called boundary points and comprise
the boundary, D, of D. Coordinate systems of type (c) restricted to D
overlap differentiably and yield a manifold structure of dimension n -- 1
on D, making D into an imbedded (n -- 1) dimensional submanifold
of M.
Let m ¢ D, and let v ¢ M,. We call v an outer vector to D if there is a
smooth curve o(t) in M with (0) m v and with o(t)  D for 0 < t < e, for
some e > 0. The orientation on M induces an orientation on i)D as follows.
Let v be an outer vector to i)D at m, and let v I ..... v,_l be a basis of the
tangent space (D),. Then we define v, ..... v,_l to be an oriented basis
of (D), if and only if v, v, .... , v,_l is an oriented basis of M,. One can
easily check that this definition is independent of the outer vector v chosen,
and that this defines a smooth orientation on i)D in the sense of 4.1.
Now let ¢o be an n-form (n -- dim M) with compact support, and let D
be a regular domain in M. As a particular case, D could be all of M. We
are going to define the integral of ¢o over D. As in 4.6, it will be sufficient for
the purposes of this definition for o to be a continuous n-form. We shall
use a partition of unity to reduce the support of o to certain n-simplices in
M over which we can integrate as in 4.6(9). First, we shall choose some n-
simplices suitably related to D and D.
An n-simplex ¢ in M will be called regular if ¢ extends to a diffeomorphism
on a neighborhood of A-. When speaking of regular n-simplices, we shall
always assume that they have been extended in this way to a neighborhood
of A-. An oriented regular n-simplex is one in which the map ¢ preserves
orientations. (We always take the standard orientation on
Associated with a given regular domain D, we shall consider only oriented
regular n-simplices of the following two types:
(oQ ¢(A n) = Int(D).
( ¢(A') = D and ¢(A n)  D m ¢(A'-); that is, precisely the
nth face of ¢ lies in the boundary of D.
Now cover D by open sets U of the following types:
(0') U lies in the interior of an oriented regular n-simplex ¢ of type
(/3') U is the image under a type ( oriented regular n-simplex ¢ of an
open set V in IP.  which is a neighborhood of a point in the nth
face of A', which intersects the boundary of A" only in that
nth face, and whose image under ¢ is contained in ¢(A')

IntaBratlon on Manolda
Since supp ¢o  D is compact, it has a finite cover UI, .... Ut by open
sets of type (') or (fl'). Let the associated oriented regular n-simplices be
oI ..... o. Let U -- M -- (supp ¢o  D), and let q, qh, .... q be a
partition of unity subordinate to the cover U, UI,..., Ut of M. We define
the integral of ¢o over D by
f:
We must check that th definition (I) is independent of the cover and the
partition of unity chosen. Let V, VI ..... V and W, Wl,-.-, W be another
such cover and another such partition of unity respectively, with V associated
with the oriented regular n-simplex -. Since  = 0 on supp ¢o  D, it
I
follows that    1 there, so that
,U.,,,o ':., f:
(2) - :Z :Z:¢o - :Z ,,,.
Similarly,
(3)
Now, since a -1 o  is an orientation-preserving diffeomorphism on the open
set (possibly empty) where it is defined, and since (supp qdo)  a(A) _--
(supp 0q¢o) (A), it follows from the change of variables formula
4.5(2) that
It follows from (2), (3), and (4) that . ¢o is well-defined, independent of the
choice of cover and partition of unity.

148 Integration on Manolds
Observe that if 7' is a diffeomorphism of M, then
(5)
with "-I-" if and only if y is orientation-preserving.
We now are reacty to state and prove our second version of Stokes'
4.9 Stokes' Theorem II Let D be a regular domain in an oriented
n-dimensional numO'old M, and let w be a smooth (n -- 1) form of compact
npport. Then
tntooF Let ,..., q and el,..., u be chosen as in 4.8(1) relative
to (aupp w)  D. Since  q -- 1 on a neighborhood of (supp w)  D,
d( q) -- 0 there. Thus on a neighborhood of (supp w)  D we have
(2)  d() -  d, ^  +  , d - d.
--1 --1 --1
Now, if ¢ is an n-simplex of type 4.8(a), then
t D
since supp q,w  Int ¢(A )  Int D. On the other hand, suppose that
¢ is an n-simplex oftype 4.8(). In this case, qo is zero on the boundary
of ¢ except possibly at points in the interior of the nth face o . Now
¢ is an orientation-preserving regular (n -- 1) simplex in D ifn is even,
and is orientation-reversing if n is odd. It follows that
From (2), (3), (4), and from our first version (4.7) of Stokes' theorem,
we see that
(5)
Let ¢o be a smooth (n -- l) form on a compact oriented n-dlmen-
Coronary
sianal manifold M. Then

ltrato on aold$ 
4.10 Integration on a Riemannian Manifold Let M be a Rieman-
nlan manifold of dimension n. That is, M is an n-dimensional differ-
entiable manifold with a positive definite inner product  , ), on ench
tangent space M. such that m - X, Y). is a smooth function on M when-
ever X and Y are smooth vector fields. The existence of Riemannian metrics
on differentiable manfolds was asserted in Exercise 23 of Chapter I.
Given a point m ¢ M, one can find a neighborhood U of m and a collec-
tion el, • .., e. of C ® vector fields on U which are orthonormal in the sense
that they form an orthonormal basis of the tangent space to M at ench
point of U. Start with a coordinate neighborhood (U, xl .... , x.), apply
the usual Gram-Schmidt procedure to orthonormalize the vector fields
8/Sx .... ,8/Sx., and do it simultaneously at all points of U. Such a
collection e .... , e is called a local orthonormal frame field.
Since the inner product  , ),. is, in particular, a non-singular pairing
of M., with itself, it induces (see 2.7) a natural isomorphism of M.-with
M*., namely, v - qo where
Via this isomorphism, M inherits an inner product. Observe that the dual
basis to an orthonormal bas/s of M, is itself an orthonormal basis for M*.
Now let e ..... e be a local orthonormal frame field on U, and let
co,... , co be the dual 1-forms. That is,
(2) co,(e) , . on U.
Then co ..... co form a local ortononnal coframe field on U. Consider
now two local orthonormal coframe fields col ..... co on U and co ..... co,
on U'. Then on U  U'
t
(3) col ^"" ^ co. = det(o)co ^..- ^
where o is an orthogunal matrLx whose entries are C ® functions on U  U'.
Thus
Now assume that M is oriented. A local coframe field co ..... co,,
on U will be called oriented if co ^ • • • ^ co belongs to the orientation at
each point of U. Choose a local oriented orthonormal coframe field about
each point of M. Then the corresponding n-forms co ^ • • • ^ co,, agree on
overlaps, and therefore determine a globally defined nowhere-vanishing
n-form co on M. This form co is called the volume form of the oriented
Riemannian manifold M. Its integral over M is the volume of M.
In Exercise 13 of Chapter 2, the star operator • was introduced on
A(V) for an oriented inner product space V. On an oriented Riemannian
manifold M, we therefore have • defined on A(M*) for each m. It is easy
to see that • takes smooth forms to smooth forms, so we have a linear
operator
(5) *: E'(9 -

150 Integration on Manifolds
which, according to Exercise 13 of Chapter 2, satisfies
(6) ** ffi (-l),C'-,.
We already know from 4.8 how to integrate n-forms over an oriented
n-dimensional manifold. Now in the case of an oriented Riemannian mani-
fold M, we define the integral over M of a continuous functionf with compact
support to be the integral of the (continuous)n-form .f--rio. That is,
(7) f . fx ,f . fx
Actually, the orientation was convenient but not necessary in the defini-
tion of J'tf. If M is a (not necessarily oriented) Riemannian manifold,
we define the integral of continuous functions with compact support as
follows. Let {U,} be a cover of M by interiors of regular n-simplices
and let w," ..... w," be a local orthonormal coframe field defined on a
neighborhood of o.(As). Then there exist C ® functions h, on neighborhoods
of A s such that
¢{o.(¢o," A • " A cos" ) , h. dr, A • .. A dr..
Let {%} be a partition of unity subordinate to the cover {U,}, and letf
a nfinuom funon wi mpt sup on M. en we define
That is deflation is indendent of the cover and paition of uni chon
follows from an arment milar to the one at the end of 4.8. In the  of
an oent emannian manifold, (8) and (7) a.
In classic vtor alysis the ent of a function f on " is defined
to  e rotor field  (8Sr)8/Sr, and e diveron of a vtor field
V  v [r, is defin to  the function  Ov,/Or. We extd e
notions to ne emannian mifol  follows. R tt the metc
Ov us oni isomohisms M,  M. We shall for conveen
deno such isomohisms by a tilde, so that if v  M, then  will  the
nng dl element in M; and if w  M, then  will  the
soding vtor in M,. Then iffis a function on M, its aent is the tor
field
(9) df ffi d
If Visa vtor field on an ofient Riemannian manifold, then its divergen
is e funion
(10) dlv V ffi ,d,.

Integration on Manifolda
Stokes' theorem has an equivalent version on oriented Rienumnian
manifolds known as the divergence theorem. This theorem says that if
V is a smooth vector field on an oriented Riemannian manifold M, if D ta a
regular domain in M, and If  is the unit outer normal vector field on )D,
then
(II)
The proof is left to the reader as an exercise. (For a few additional remarks
see Exercise 4.)
4.11 Integration on a Lie Group Let G be an n-dimensional Lie
group. We observed in 4.3(a) that {7 is orientable. We now fix once and for
all an orientation on G.
Consider the left invariant n-forms on G. Since such a form is uniquely
determined by its value at one point, and since the nth exterior power of an
n-dimensional vector space is one-dimensional, there is exactly a one-
dimensional space of left invariant n-forms on G. Choose a non-zero left
invariant n-form o consistent with the fixed orientation on G.
Since G is oriented, the integral of compactly supported n-forms is defined
on G as in 4.8. We now define, with respect to o, the integral of a compactly
supported continuous functionfon G by setting
The integral (1) depends, of course, on the choice of the non-zero
left invariant n-form o consistent with the orientation on G. But since such
forms are uniquely determined up to a positive constant multiple, so is the
integral (1). In the case of a compact group G, we can and always will
fix the choice of o by requiring the normalization
(2) f ----
I.
Consider the diffeomorphism I., which is left translation by the element
r of G. Then since 61.(o)----o, I. is orientation-preserving, so that,
according to 4.8(5),
In view of property (3)---that the integral of a functionf on G is the same
as the integral of any of its left translates fo/.--we call the integral (I)
left Invariant.

152 Integration on
Now we ask to what extent the integral (I) is also right invariant. That is,
when do we have
for each  ¢ G? The form row is still left invariant, since
Thus SroO is some constant multiple of o. Thus there is detined a function
[ of G into the non-zro ri numbers such that
(5) r.() - [(a.
It is easily checked that [ is C ®. We let
(6) () = I()I.
Observe that
(7) () -- ()(),
so that  is a Lie group homomorphism of G into the multiplicative group of
positive real numbers.  is called the modular function. Now since, by
4.8(5), for each r in G
it follows that the integral (1) is right invariant if and only if  = 1 on G.
A Lie group G for which  = 1 is called unimodular. We observe that
each compact Lie group G is unimodular since for ch a ¢ G
Thus the integral on a compact Lie group is both left and rightinvariant.
4.12 Application of 4.11 A typical application of the integral on
a compact Lie group is the following.
Let G be a Lie group, and let a: G--* Aut(V) be a representation into
the. automorphisms of a real or complex inner product space V. The
representation a is called unitary (respectively orthogonal) in the case in
which V is a complex (respectively real) inner product space if
(I) <()v, ()w) - <v,w)
for all v and w in V and for all r s G.
Let G be compact and V complex (respectively real). Then there Is an
inner product on V with respect to which  ts unitary (respectively orthogonal).
The proofs in the real case and in the complex cases are similar. Let { , }
be any inner product on V. We set

de lbrn Cohomology 153
(2)
where we use d to denote that we are considering the integrand as a function
of o in G. It is/mmediate that ( , ) is again an inner product. That (I)
holds follows from the right invariance of the integral on G:
DE RtIAM COHOMOLOGY
4.13 Definition A F-form  on a diffrentiable manifold M is called
closed ifd0 -= 0. It is called exact if there is a  -- 1) form/ such that  *= d/.
Since d* *= 0, every exact form is closed. The quotient space of the real
vector space of closed F-forms modulo the subspace of exact F-forms is
called the pth de  cohomology group of M.
O)
H.a(M) == {closed p-forms)/{exact p-forms).
4.14 Example Consider the case of the unit circle 5% Since there are
no non-zero p-forms on 5  for p > 1, all of the cohomology groups He s(5 )
are zero except possibly for p ---- 0, 1. There are no exact 0-forms, and a
closed 0-form on a connected manifold is s/reply a constant function, so
O) HL  (S
The "polar coordinate function" O on S  is not well-defined globally since
it is defined only up to integral multiples of 2=. However, its differential
dO is a globally well-defined nowhere-vanishing l-form on S . In fact, dO
is the volume form of the natural Riemannian metric which 3 " inherits from
lt. Now, dO is not exact, for if it were, its integral over 3 " would have to
be 0 rather than 2=. All l-forms on S I are closed. We claim that if = is
a l-form, then there is a constant c such that o -- c dO is exact. For let
 ,=f(O) dO, let
C
and let
g(O) ==Jo (f(O) -- c)
dO.
Since g(O + 2=n)
function on $1; and dg ---- (f(O) -- c)dO , o -- c dO. Thus every l-form
on S  differs from a real multiple of dO by an exact form. Consequently,
(2)

4.15 Effect of Mappings Let f: M-,. N be a C" map. Then the
algebra homomorphism /}f: E*(N)--* E*(M) commutes with d, according.
to 2.23, and hence maps closed forms to closed forms and exact forms to
exact forms. Thus it induces a homomorphism
(1) f*: H. R(N) -- Ho x(M)
for each integer p  0. If, in addition, g: N-,. X is C ®, then
(2) (g of)* -ffi f* o g*.
Clearly the identity map id: M-M induces the identity on de Rham
cohomology:
(3) (id)* == id.
It follows from (2) and (3) that a diffeomorphism f: M--* N induces iso-
morphisms on de Rham cohomology. Thus the de Rham cohomology is a
differentiable invariant of a differentiable manifold M. We shall prove in
Chapter 5 that it is actually a topological invariant. That is, the de Rham
cohomology groups depend only on the underlying topological structure of
M and do not depend on the differentiable structure. A key part in the proof
of this fact is the de Rham theorem, a version of which we shah formulate
now. First, we need to define the real differentiable singular homology
groups of M.
4.16 Real Differentiable Singular Homology For each integer p  0
we let =S,(M,I) denote the real vector space generated by the differentiable
singular p-simplices in M. Hence the elements of ®S,(M, ) are precisely the
differentiable singular p-chains in M with real coefficients. For p < 0,
we let =S,(M,) be the zero vector space. The boundary operator a induces
linear transformations
(1) a,: **S,(M,)--*
for each integer p, which for p < 0 are simply the zero transformation.
According to 4.6(7), ,. +1 ffi O, so that the image of  lies in the kernel
of . Thepth differential singular homology group of M with real coecients
is defined by
(2) ®H,(M;) =ffi ker a,/Im 8,
and is moreover a real vector space. Elements ofker 8, are called differentiable
p-cycles, and elements of Im 8 are called differentiable p-boundaries.
4.17 The deRhamTheorem We shall define a linear mapping
of the de Rham cohomology H, foR(M) into the dual space =H,(M;I)*
of the real differentiable singular homology:
(1) HI° R(M) --, ®n,(M;n)*.

de Rham Cohomology
Let = be a closed ffform representing the d Rham cohomology class {¢},
and let z be a p-cycle representing the real differentiable singular homology
class (z). Then (I) is defined by
(2) {¢}({z}) ,= f .
That (2) is independent of the re,oresentatives  and z chosen follows im-
mediately from Stokes' theorem I (4.7).
The de Rham theorem asserts that (1) is an isomorphism. This will be
proved in 5.36 and 5.37. The real numbers deterrained by the integrals of a
differential form over differentiable cycles are called the periods of the
differential form. Now, Stokes' theorem I says that the periods of an
exact form are all zero. The injectiveness of the isomorphism (1) gives a
converse, namely, if a closed form has all of its periods zero, then it is
an exact form. The surjectivity of (1) says that if a real number per(z) is
assigned to each cycle z in such a way that
(3) per(azl + z2) ---- a per(zO + per(zs), per(boundary) ,= 0,
then there is a closed form  on M such that for all cycles z
(4) f, ffi per(z).
A key ingredient in the proof of the de Rham theorem (see 5.28) is the
4.18 Poincar[ Lemma Let U be the open unit ball in Euclidean space
", and let Et(U), as usual, be the space of differential k-forms on U. Then
for each k  1 there is a linear transformation ht: Et(U)--, E-(U) such that
(1)
h+ o d -l- d o h ffi= id.
PROOF We begin with formula 2.25(d), which expresses the Lie
derivative in terms of exterior differentiation and interior multiplication:
(2) L:¢ =ffi i(X) o a -l- a o i(x).
we shall apply (2) to the radial vctor field
(3) X ----,.r, r 
on U. We define a linear operator  on E(U) by setting
(4) (J'drq ^... ^ drk)(p)  t-lf(tp) dt dry1 ^ ... ^ dr(p)
and extending linearly to NI f (U). N ith  defined by (3),
we show that
(5) t o L: = id on EL(U).

I
!56 Integratlon on Man'olds
For, using the fact that L x is a derivation which commutes with d,
(6) t ° L(f dr. ^'" ^ dr,.)(p)
-- (f:P-'(kf(tP) ÷ Y-r(tP) 8-rL) dt) drq ^'" ^dr,.(P)
- (fOo)) d d,, ^... ^
-- f,) dr,, ^... ^
From (5) and (2) we obta/n
(7) id =, a o I(X) o d 4- a o d o l(X)
on Et(U) with X given b.y (3). Now a commutes with d; that is,
(8) a. od-- do a_.
For
a o d(f dr ^'" ^ dr.X p)
-- a,(,rdr ^ dr ^" " ^ dr.)(p)
- d(f;y(t,)dr)
 from (8) d if) we obn
(9) id -- ..i(.d + d.: oi(
on E(. us ¢ d 1i ffoafion h wch yids (1)
is obta by
 (a) If   a k-form, k  1, on t open t ball in " 
-- O, t tre ext a (k -- l) fo  (nely h(w)) ch tt dO -- .
ro )  de m comlogy gr of t open it ll 
e all zero for p  1.

Exercises 157
4.19 Renmrk Letf andf be C" maps of M into N. Then we have
induced maps
(I) Of: E(N)--E(M) for each/.
If we w/sh to prove that f and f, both induce the same homomorphism
on de Rham cohomology, then we need only find a collection of I/near
transformations
(2)
such that
(3) h,+1 od + do, =  -- f,.
For then, i/" z is a closed form on N, (z) and fs(z) differ by an exact
form. and thus lie in the same cohomology class. Such a collection of I/near
transformations () is called a omotopy operntor for f and fs. In the
Poincar lemma we found a homotopy operator between the identity map
of the open unit ball U and any constant map (range I point) of U into itself.
Prove the assertion of 4.3(c) that a d-dimensional manifold X for
which there exists an immersion f: X-* IR,+ is orientable if and
only if there is a smooth nowhere-vanishing normal vector field
along (X,f).
Prove that the real projective space P" is orientable if and only if
n is odd. (Hint: Observe that the antipodal map on the n-sphere
$" is orientation-preserv/ng if and only if n is odd.)
Carry out in detail the proof of the existence of local orthonormal
frame fields on a Riemannian manifold.
4
Prove the divergence theorem 4.10(II). This follows from Stokes"
theorem together with the identity
(1) fz,  ,= f V,r).
The easiest way to see (1) is to choose a local oriented orthonormal
frame field e ..... e. on a neighborhood of a point of iD, such that
at points of iD, e is the outer unit normal vector and e, ..... e.
form an oriented basis of the tangent space to /D. Then express
• I 7 and  V,r) in terms of this local frame field and its dual coframe
field ¢o .... , o,.

158 Integration on Man'old
Let M be an oriented Riemannian manifold, let f and g be
functions on M, and let D be a regular domain in M. The Laplaian
of g, denoted Ag, is defined by
(1) Ag . --*d*dg.
(For more on the Laplacian, see Chapter 6. Observe that our
choice of sign for the Laplacian yields Ag. -- _ O*g]Or* for g
a C = function on Euclidean space 11" with its standard Riemannian
structure in which {0/Or} is an orthonormal basis of each tangent
space.) If  is the unit outer normal vector field along OD, we let
Og]On denote (g). Prove the following two Green's identities:
Green's Ist: f--fgradf, gradg)--ffAg.
6 Let o be the volume form of an oriented Riernannian manifold of
dimension n. Let X, .... , X, and ¥, .... , Y be vector fields on
M. Prove that
o(X, .... , X,,) " o(Y, .... , y,,) R
Prove also that
.... , - ^--. ^
where ' is the l-form dual (via the Riemannian structure) to the
vector field X.
7 Prove the differentiability of the function [ of 4.11(5).
8 Let G be a compact (oriented) Lie group, and let (o) -- o --x for
¢,  G. Prove that for every continuous functionfon G
f  f-f  fo..
9 Prove that a Lie group G has a bi-invariant Riemannian metric
(that is, a metr/c such that both all. and dr. preserve the inner prod-
ucts for each ¢,  G) if and only if the closure of Ad(G) in Aut(g)
is compact.
10 (a) Prove that H{e R(") R 0 for each p  I and each n  I.
(b) Prove that Ho . R(M) ---  for a connected manifold M.
11 Determine the de Rham cohomology of the annular region
1 < (r; + r,')'/' < 2 in

Exercises 19
12 If = and fl are closed differential forms, prove that = ^ ,8 is closed.
If, in addition, fl is exact, prove that = ^ fl is exact.
13 Consider the l-form = = (x* + 7y) dr + (--x + y sin)) dy on IRa.
Compute its integral over the following l-cycle z.
0,o)
14 Let = *= (2x + y cos xy) dr + (x cos xy)
is closed. Show that = is exact by finding a function f: IRa--,
with = m dr. What would the integral of = over the cycle of Exercise
13 be?
IS Let I x dy -- y dx
2r x* +
Prove that = is a closed l-form on IR' -- {0}. Compute the integral
of at over the unit circle S 1. How does this result show that = is
not exact? How does this show that /i(=) is not exact, where
i: S  --* IR' is the canonical imbedding?
16 (a) Prove that every closed 1-form on S t is exact.
(b) Let
rl dr, ^ drs -- r, dr ^ drs -l- rs dr ^ drs
(rx* + rs* +
in IR* - {0}. Prove that r is closed.
(c) Evaluate j's,r. How does this show that r is not exact?
(d) Let
r dr 4- rs des 4- • • •
(r s + rsS +... +
in IR" -- {0}. Find ,at, and prove that *at is closed.
(e) Evaluate j's,_,at. Is ,at exact?
\17 Using de Rham cohomology, prove that the torus T  is not diffeo-
" morphic with the 2-sphere S s.

160 Integration on Manf olda
18
19
(a) Prove that every closed 1-form in the open shell
in I s is exit.
(b) Find a 2-form in the nve shell nt is cl but not ex.
(c) Prove nt e nve shell is not diffeomoc wi e on
unit ball in .
t f and g  C  maps of M into N which  C  homotopic;
that is, e exis • C  map F of M x (--, 1  ) into N, for
some •  0, such that F(m,0)f(m) and F(m,l)g(m) for
eve m  M. Prove that the indu homomohismsf* and.g* of
 (N) into H (M)  equ for eh intent p. (Hint: You
will nd to prove that e two injtio i(m) (m,0) nd
i(m)  (re,l) of M into M x (--, 1  ) indu the same hom
mosms on de nm cohomolo. To prove this, find suitable
homotopy oto. The oune of the proof of the Poincnr
lemmn 4.18 should  helpful.)
() t f: M"  '+  an immeion, and let M   ven e
indu eninn sctu; nt is, for mM and
Su  M is orient, d nt a is e oriented unit
normal field along f(M'). is mes nt a,d),... ,
df(,) is   an orient ohonormn! bis of the tannt
sp to Euclide spe at f(m) whenever  ..... , is an
oriented ohononl bis of M.) Show nt the volume
form on M is given by
(b) t D  an o set in the xy ple, and let : D  
 • smooth map of the form
us  deteines an imded su in . Give D and
 the standa orientations. Use pn () to prove that the
induced volume fo on D is given by

The principal obJective in this chapter is a proof of the de Rham theorem,
one version of which we have stated in 4.17. In its most complete form it
asserts that the homomorphism from the de Rham cohomology ring to the
differentiable singular cohomology ring given by integration of closed forms
over differentiable singular cycles is a ring isomorphism. The approach will
be to exhibit both the de Rham cohomology and the differentiable singular
cohomology as special cases of sheaf cohomology and to use a basic unique-
hess theorem for homomorphisms of sheaf cohomology theories to prove
that the natural homomorphism between the de Rham and differentiable
singular theories is an isomorphism. As an added dividend of this approach
we shall also obtain the existence of canonical isomorphisms of the de Rham
and differentiable singular cohomology theories with the continuous singular
theory, the Alexander-Spanier theory, and the (ech cohomology theory for
differentiable manifolds. From these isomorphisms we shall conclude that
the de Rham cohomology theory is a topological invariant of a differentiable
manifold.
No previous knowledge of sheaf theory nor of any of these cohomology
theories is assumed. We shall develop those aspects of the theories necessary
for our applications. We begin with an introduction to the theory of sheaves
and sheaf cohomology. Our approach is based largely on the development
given by Cartan in [4]. The interested reader will find more general treat-
ments of sheaf theory in Caftan [4], Godement [7], and Bredon [3]. Sheaves
are a very powerful tool in the study of complex manifolds. For an excellent
account of their use in the theory of Riemann surfaces, see Gunning [8].
Let M be a difl'erentiable manifold. Recall that M is assumed to be second
countable and is therefore/aracom/act. It should be pointed out that nearly
all of the sheaf theory presented in this chapter depends only on M being a
paracompact Hausdorff space. The differentiable structure on M will be
invoked only in a few examples and in the discussions of the de Rham
cohomology (5.28) and the differentiable singular cohomology (5.31); and,
in addition, the locally Euclidean structure will be invoked in the discussion
of the singular cohomology (5.31). Otherwise, M need only be a paracompact
Hausdorff space.
162

Sheaves andPresheaves 163
Throughout this chapter, K will be a fixed principal ideal domain. We
shall treat sheaves of K-modules over M. The most important cases to keep
in mind are these:
(i) K is the ring of integers Z, whence a K-module is simply an abelian
group.
(ii) Kis the field of real numbers IR, whence a K-module is simply a real
vector space.
For the first few sections of this chapter, K could be any ring. The additional
property that K is a principal ideal domain will be used first in 5.13.
SHEAVES AND PRESHEAVES
5.1 Definitions A sheaf $ of K-modules over M consists of a topological
space $ together with a map -: $ -* M satisfying:
(a) - is a local homeomorphism of $ onto M.
(b) --X(m) is a K-module for each m ¢ M.
(c) The composition laws are continuous in the topology on $.
We elaborate on (c). Let $ o $ be the subspace of $ × $ consisting of all
pairs (sx,ss) such that '(sl)----'(ss). Then (c) requires that the map
(sx,ss) -* sx -- s of $ o $ -,- $ be continuous, and that each of the maps
s  ks of 8-- $ for k  K be continuous. It follows easily that the
maps s  (--s) and (s 1,s2)  sl + s2 also are continuous. Sheaves of K
algebras are defined similarly, with the additional requirement in (c) that
the map (sl ,ss)  s. ss of $ o $ -- $ be continuous.
The map r is called the projection; and the K-module 8, == r-X(m) is
called the stalk over m  M. Let U c M be open. A continuous map
f: U --* $ such that r of == id is called a section of $ over U. The O-section
is the section which associates with m  U the zero element of $,. We let
I'($,U) lenote the set of sections of $ over U. We make I'($,U) into a
K-module as follows. Let f and g belong to I'($,U), and let k  K. We
define the sum off and g to be the section
(f + g)(m) ,= f(m) + g(m) (m  U),
and we define a section kfby setting
(k f)(m) = k(/(m)) (m  U).
With these operations, I'($,U) becomes a K-module. The module of sections
of $ over M (global sections) will simply be denoted by I'($). Observe that
since - is a local homeomorphism, sections are open maps; and if sections f
and g agree at m  M, then they must agree on a neighborhood of m.

164 Sheaves, Cohomolog, and the de Rham Theorem
$.2 Examples The most elementary example of a sheaf over M is that
of a so-called constant sheaf  -- M × G, where G is a K.module with the
discrete topology and ff is given the product topology. Here the projection
is simply r(m,g) -- m.
A less trivial example is the following. Let e,, denote the set of germs of
C ® functions at m  M (as in 1.13), and let
(l) ¢®(SO -- U ,e..
When it will introduce no confusion, we shall denote ®(M) simply by ®.
We define the projection r: ® -- M in the obvious fashion so that f  P-(o-
Associate with each open set U in M and each C ® function f on U the set
U f= : ¢ ® ,
where f,, is the germ of fat m. The collection of these sets forms a basis for
a topology on '® which makes '® into a sheaf of real vector spaces (in
fact, a sheaf of algebras since each P,, has an algebra structure). '®(M)
is called the heaf of germs afC ® functions oa M. Similarly, one constructs
the sheaf 'P(M) of germs of functions of class C • on M for each integer
, :> 0.
$.$ Remark One should be cautioned that sheaves are generally not
Hausdorff spaces. A simple example is provided by the sheaf (IR) of
germs of continuous functions on the real line. Let g(t) == 0, let.f (t) = 0
for t  0, and let f(0 ,= r for t > 0. Then the germs ofJ'and g at the origin
are distinct elements of (IR); however, .f and g have the same germ for
each t < 0. It follows that the germs of.fandg at the origin cannot be separ-
ated by disjoint open sets in r6°a(IR).
5.4 Definitions Let 8 and 8' be sheaves on M with projections sr and
r' respectively. A continuous map ?: $ --* 8' such that r' o ? = r is called
a sheaf mapping. Observe that sheaf mappings are necessarily local homeo-
morphisms, and they map stalks into stalks. A sheaf mapping ? which is a
homomorphism (of K-modules) on each stalk is called a sheafhomomorphism.
A sheaf isomorphism is a sheaf homomorphism with an inverse which is
also a sheaf homomorphism.
An open set . in the sheaf 8 such that the subset .,,  . ¢3 8,. is a
submodule of 8,, for each m  M is called a subsheafof 8. It is clear that a
subsheaf of 8, with the natural topology and the projection map which it
inherits from 8, is again a sheaf.
Let ?: $ -*." be a homomorphism of sheaves. The kernel of ? (ker ?)
is the subset of $ which maps under ? into the O-section of.', and is in fact
a subsheaf of $. The subsheaf ?($)  ." is called the image of ? (ira ?).
In the case in which q is one-to-one, we shall often identify 8 with ?(8)
and speak of the subsheaf $ of .'.

Sheaves and Preheaes 16
Let  be a subsheaf of 8. For each m  M, let .9 r. denote the quotient
module 8,,,lt,,,, and let
O) " = O ...
I.¢t ¢: $-. be the natural map which associates with each element of
$,. its coset in '., for each m, and give . the quotient topology. That is,
a set Uin ." will be open ifand only if --I(U) is open in $. "/'hen, with this
topology and with the natural projection which maps each element of
to m, . is a sheaf over M and ¢: $ - . is a sheaf homomorphism. We
leave the details as an exercise..9 r is called the quotient sheaf of g modulo
If ?: g-.9 r is a homomorphism of sheaves, then it is easily checked
that the natural map g/ker ? - im ? given by (s + (ker ? [ g,)) -, ?(s) for
s  g,, is a sheaf isomorphism.
A sequence of sheaves and homomorphi.sms
(2) ---  $  $+1  $+, ...
is called exact if at each stage the image of a given homomorphism is the
kernel of the next. Exact sequences of K-modules are defined similarly.
Observe that (2) is exact if and only if for each m  M the induced sequence
of homomorphisms of the stalks over m, namely,
(3) ... (s,). (s,+,). (s,+,).
is exact. If  is a subsheaf of $, and .9 r is the quotient sheaf $/t, then the
natural sequence of homomorphisms
(4) 0- - 8-.9r-- 0
is exact, where 0 denotes the constant sheaf over M whose stalk over each
point is the trivial K-module consisting of only one element. Exact sequences
of the form (4) consisting of only five terms with the first and last being
the 0 sheaf are called short exact sequences. Short exact sequences of
K-modules are defined similarly.
5.5 Presheaves A presheaf P ,= {S o ;Pry.v} of K-modules on M consists
of a K-module Sry for each open set U in M and a homomorphism
Pry.v: Sv " Sry for each inclusion U  V of open sets in M, such that
pry.ry == id, and such that whenever U  V  W, the following diagram
commutes:

166 Sheaves, Cohomolog),, and the de Rhmn Theorem
A typical example of a presheaf is the following. Associate with each
open set U in M the algebra C®(U) of C ® functions on U, and let Pv.v
be the map which restricts a C ® function on V to a C ® function on U.
This yields a presheaf {C®(U);pr/.v} of real vector spaces (in fact algelras)
on M. Similarly, one could associate with U the vector space of p-forms
on U fo some fixed p and again let Pv.v be the restriction mapping.
Let P {Sv;pv.v } and P'--"S ....
-- t v,Pv.v be presheaves on M. By a
presheaf homomorphlsm of P to P' we mean a collection {?v} of homomor-
phisms ?v: Sv -* $ such that
(2)
P,V' o PV' m pU o p
whenever U  V. A presheaf isomorphism is a presheaf homomorphism
{?r} in which each ?r is an isomorphism of K-modules..
5.6 The Relationship between Sheaves and preshenvea Each sheaf
$ gives rise canonically to a preshcaf {F($,U);pv.r}. Here we associate
with each open set U in M the/C-module I'($,U) of sections of $ over U,
and to each inclusion U  V of open sets in M we associate the homo-
morphism
(1) pv.v:
which maps a section of 8 over V to its restriction to U. We shall denote
this map of sheaves to presheaves by o. We call 0,(8) the presheaf of sections
of the sheafS.
Conversely, we shall now show that each presheaf canonically determines
a sheaf, which we call its associated sheaf. In practice, many sheaves will
in this manner arise naturally from presheaves. The best example to keep
in mind during the following construction is the presheaf {C®(U);pu.v};
its associated sheaf will be the sheaf of germs of C ® functions on M.
Let P -- {$v;Pv.v} be a presheaf of /C-modules on M. Let m ¢ M,
and let 5", be the disjoint union of each of the modules Sv for which m ¢ U.
If we setf¢ S v equivalent to g ¢ Sv if and only if there is a neighborhood W
ofm with W  U c V such that pw.vf= Pw.vg, we obtain an equivalence
relation on S*,. The set 8, of equivalence classes of elements of S*, will
be the stalk of the associated sheaf over m. If m
be the natural projection which assigns to each element of Sv its equivalence
class. Let
respectively. There exists a neighborhood W of m such that W  U c V.
Define addition in 8, by setting
3) sx + sz
and define multiplication by k ¢/ by setting
(4)

It is easy to check that the operations in (3) and (4) s well-de, fined and give
$.. the structure of a K-module such that the maps P...v are all homo-
morphisms. $.. is known as the direct limit of the modules Su for U
containing m. Now let
(5) S-- U S=,
and let -: g--+ M be the obvious projection such that -($) -- m. We
topo]ogize $ by taking for a basis of the topology the collection of subsets
of $ of the form
(6) Ol ' {P,. uf: P ¢ U}
.for the various f¢ Su and the various open sets U in M. This collection
does indeed form a basis for a topology on $, for if s e Ol  O, say s
p,.uf "= P,.rg, then there is a neighborhood W of p with W c U  V
for which pw.uf ,= pw.vg, from which it follows that s e Op,,.,, c O,  O.
We claim that $ is a sheaf over M with projection -. Now - is a local
homeomorphism since it is a homeomorphism on each O/. Each stalk
has been given the structure of a K-module, so finally we need only check
that the composition operations axe continuous. Consider first multiplica-
tion by k e K. Let s e $, and let O be an open neighborhood of ks. Let
s ,= P,.vf Then Or/is also an open neighborhood of ks. Let V be a neigh-
borhood ofp contained in -(Ot I  O). Then Opt /is an open neighborhood
of s which maps under multiplication by k into the open neighborhood
O,..rt/c O of ks. Finally, we show that the map (sx,s)-+sx- sz of
$ o $ -* $ is continuous. For let Ol be an open neighborhood of sx -- sz,
say sx -- s := P.uf, and let sx =: P.r'g and s =: p.wh. Then there exists
an open neighborhood ( ofp with ( c U  V  W such that
It follows that the open neighborhood
(8) O0.r o x O., f g o $
of (s, s) maps into Or. Thus the composition operations are continuous,
and $ is a sheaf over M with projection *-. $ is called the sheaf associated
with the presheaf P. We shall denote this map of presheaves to sheaves by
/; thus $ ffi/(/').
Clearly, each sheaf homomorphism p: $ --* f gives rise to a presheaf
homomorphism between the presheaves of sections ($) and (-) by com-
posing elements of F($,U) with p. Conversely, a presheaf homomorphism
{Pu} from P to P' canonically induces a homomorphism p of the associated
sheaves/(P) and/(P') such that
(9) , p o
p.U o ¢Pu -- P.u

168 Sheaves, Cohomology, and the de Rham Theorem
for each open set Uin Mand eachp  U. Moreover, in going in either direc-
tion, from homomorphisms of sheaves to homomorphisms of presheaves,
or vice versa, the composition of two homomorphisms induces the composi-
tion of the corresponding homomorphisms. 
Suppose now that we start with a sheaf g, take its presheafz(g) ofsections,
and then form the associated sheaf((g)), thus obtaining the sheaf ofgerms
of sections of g. Then the sheaves g and/((g)) are canonically isomorphic.
Indeed, let f /(=(g)) be the germ at p of some sectionf of g over an open
set U containing p; that is, f -- P,.vf for f P(S,U). Then [ -.f(p)
determines a well-defined map (o(g)) -, g which is a sheaf isomorphism.
We leave the details as an exercise.
Generally, however, if we start with a presheaf P and take the presheaf
o((P)) of sections of the sheaf associated with P, we may get a presheaf
quite distinct from P. For example, let P- {$v;PtT.v} where Sv is the
principal ideal domain K for each U, and all restrictions Pv.v for U . Vare
identically 0. Then the presheafo((P)) .assigns to each open set U in M the
zero /C-module. The conditions of the following definition are clearly
necessary if P is to be isomorphic with o((P)), and it will be shown in 5.8
that they are also suticient.
5.7 Definition A presheaf {Sv;Pv.v} on M is said to be complete if
whenever the open set U is expresod as a union U] U, of open sets in M,
the following two conditions are satisfied:
(C)  Wheneverfand g in Sv are such that Pvs.vf" Pv,.vg for all
then f-- g.
(C) Whenever there is an element f,  Sv, for each o such that
Pvsnv.vf, " Pvsnv.vf for all o and , then there exists
f Sv such that f,  pv,.vffor each o.
5.8 Proposition If P  a complete presheaf, then o((P)) ts canonically
omorphic with P.
p]tooF Let P ,-{Sv;Pv.v}. We define a presheaf homomorphism
from P to o((P)) as follows. For each open set U in M we map
by sending the elementf of Sv to the section
(2) P p,.vf
of (P) over U. To prove that this presheaf homomorphism is an
isomorphism, we need to prove that each of the homomorphisms (1)
are isomorphisms. Injectivity follows from 5.7(Cx), for if fin Sv maps
to the 0-section, then there exists a cover {Us} of Usuch that
pv,.v(O). Therefore f,- 0  Sv by (C). To prove surjectivity, let c

Sheaves and Presheaves 169
be a section ofp(P) over U. Then for eachp e U, there is a neighborhood
U, ofp and an element f, e So, such that
(3) P,. r.rf, l C(f)
for all f  U,. It follows from (3) and 5.7(C that for each p and f
in U,
Thus, according to 5.7(Cs), there existsf Sr such that
(5) f, "-
for eachp  U. It follows from (5), (3), and (2) thatfmps to c under
the homomorphism (I).
5.9 Tensor Products Before we define the tens.or products of sheaves
and presheaves, we need a few preliminary remarks. The definition of the
tensor product $ ® T of K-modules S and T is completely analogous to
that for the special case 2.1 of tensor products of R-modules. If f: S -* $'
and h: T-* T' are homomorphisms of K-modules, then their tensor product
f® h is the homomorphism of $ ® T into S' ® T' uniquely associated by
the universal mapping property 2.2(,,) with the bilinear map
(1) (s,t) -* f(s) ® h(t)
of S × T into $' ® T'.
Observe that if $ is a K-module, then there is a canonical isomorphism
(2) s®g $.
Indeed, let f: S ® K-,. S be the homomorphism determined by the bilinear
map (s,k)  ks ofS × K-* S. Thenfis clearly surjective, and is also injec-
tive since (s®k)=(ks)®l, so if f(s®k))=O, then
 ks = O, which implies  (s ® k) == 0.
Now let P = {Sv;pv.v} and P'= {Sr,pv.v} be presheaves on M.
Then their tensor product is the presheaf
(3) P ® P' = {$v ® $; ,v.v ® ,'v.r}.
]f {?v}: P-'* Q and {?r}: P'-*" Q' are homomorphisms of presheaves
(5.5(2)), then their tensor product {?v} ® {?5} is by definition the homo-
morphism {Pv ® ?r} ofP ® P' into Q ®
]f $ and .f" are sheaves over M, we define their tensor product $ ®
to be the sheaf associated with the tensor product of the presheaves of sections
of $ and .f'. That is,
(4) S ® - =/(=(S) ® =(')).

170 heaves, Coomology, and tlw de Rham Theorem
Moreover, if ?: $--* " and y: $'... "' are sheaf homomorphisms, and
if (?r]}:0(8)--*=(.') and ('r]}:=($')"*=(") are the corresponding
homomorphisms on the presheaves of sections, then we define the tensor
product ? ® ), to be the homomorphism $ ® $' ... " ® "' associated
with the presheaf homomorphism
(5) {) ® {,): (s) ® (;') --. (Y3 ® (Y").
The reader should check that there is a canonical isomorphism
(6) (S ®
so in particular if ." is the constant sheaf M x K, then in view of (2) there
is a canonical isomorphism
(7)
Moreover, if ? and ), are sheaf homomorphisms as above, then the reader
should check that
(s) ( ®r)l(s ® s'). ffi [ s.®r[ s,.
One might ak, "Why not define 8 ® " and ? ® , by (6) and (8) rather
than going to presheaves and back?" We could indeed proceed in this way,
but it would involve a considerable duplication of previous work. The
reason is that if we define the tensor products by (6) and (8), then we would
also have to define the topology on 8 ® " and check that the properties
5.1(a) and (c) are satisfied, and would also have to prove that ? ® , is
continuous. In defining $ ® " and ? ® , by going to presheaves and back,
the work of showing that $ ® " is really a sheaf and that ? ® , is a sheaf
homomorphism ha already been carried out in the analysis of the maps
 and p of 5.6.
$.10 Fine Sheave A sheaf 8 over M is said to befine iffor each locally
finite cover (U) of M by open sets there exists for each i an endomorphism
! of $ such that:
(a) supp(l,)
(b)  !, ffi id.
Here, by supp(l) (the support ofl) we mean the closure of the set of points in
M for which I [ $, is not zero. We shall call (1) a partition of unity for 8
subordinate to the cover (U) of M.
An example of a fine sheaf is the sheaf '®(M) of germs of C ® functions
on M. If (U) is a locally finite open cover of M, let (?} be a partition ofunity
on M subordinate to this cover (Theorem 1.11). We obtain endomorphisms
 of the presheaf (C®(U);pv.r,) by setting
(I) (f) ---- (?, [ U) "f forf C'°(U).
The associated sheaf endomorphisms l of '®(M) form a partition of unity
subordinate to the cover (U} of M.

Shetwes and Pr esheatms 171
Let $ and .9" be sheaves over M, with $ fine. Then $ ® .9" is itself a fine
sheaf. Indeed, if {/} is a partition of unity for $ subordinate to the cover
{Us} of M, then {l, ® id} is a partition of unity for $ ® .9".
5.11 A sheaf homomorphism ?: 8--*.9" gives rise to a homomorphism
F($) --, F(') of the modules of global sections by composing sections of
with ?. We leave as an exercise the fact that a short exact .quence
(I) 0-- 8' --,. 8--*
gives rise to an exact sequence
(2) 0 - r(s') - r(s) - r(s').
However, the homomorphism F(8)- F(8") is generally not surjective.
Consider the following example.
Let $1.s be the "skyscraper" sheaf over a connected M whose stalk
is the zero K-module over each point except over the distinct points Pl and
pz where the stalk is K. (Note that the topology on $,t.,z is uniquely deter-
mined by the requirement that $,t.,z be a sheaf.) Let f" as usual be the
constant sheaf with stalk K. There is an obvious homomorphism of
onto $1.z, namely, the homomorphism is zero on all stalks of at ° except
on those overpl and Pt .where it is the identity map. However, the associated
map F(oW)-* F($,,.,t) cannot be surjective for F(oW)--K, whereas
Exactness of (1) is a purely local property, whereas exactness of (2)is a
global property. We will see that certain sheaf cohomology modules will
provide an extension of the exact sequence (2) and thus will provide a
measure of the extent to which F($) --* F($') fails to be surjective.
5.12 Theorem Let ?: 8 --* .9" be a surjective sheaf homomorphism with
kernel t. Suppose that t is a fine sheaf. Then the homomorphism
r($) -, r(a-) is suqectit, e.
eooF Let t be a global section of '. We must construct a section s
of $ such that p o s ,= t. By the continuity of p and t and by the property
5.1(a) of $ and " there is a covering {Us} of M by open sets and, for
each i, a section st of $ over Us such that
(1)  o s, = t I v,.
Sinb¢ M is paracompact, we can assume that the cover {Us} is locally
finite. The difference
(2) s, - st - s
is a section of the kernel  over U t U, and on Us t U t Ut the
differences satisfy
(3) s, + st -- st.

172 Sheaves, Cohomolo, and th de Pam Theorem
Now, let {/) be a partition of unity for  subordinate to the cover
(U,) of M. Consider the section ! o a o of  over U,  U. Since the
support of l lies in U, we can extend the section l o ao to be a
continuous section of  over U, by defining it to be zero on points of
,- ,. Let
(4) s =  t, o s,.
Then ; is a section of  over U, and by (3) and (4) tbe difference
(5) s - s'  Y  o s, - Y  o s,  _  o s, - s,
over u  ,. Thus
(6) s,- s  s - s'
on U  U. It follows that if we set $(m) ,- ( -- )(m) for m  U,
then  is a well-defined global section of $ such that ? o a  t.
5.13 Definitions A K-module X is torsionle if' there is no non-zero
element x  X for which there exists a non-zero element/c  K such that
kx = 0. A sheaf of K-modules is said to be torionle if each stalk is a
torsionless K-module.
We shall need the following basic lemma concerning tensor products
and exact sequences. Th/s is the first of two fundamental algebraic proposi-
tions which we shall need in this chapter for which we shall not provide
proofs. (The second propos/tion is the Kunneth formula, which we shall
need in 5.42.) The proofs are rather long, are not of particular inerest for
our purposes, and would tend to obscure the main goals of the chapter.
The interested reader can find a proof of the following lemma in Spanier
[28, pp. 215, 221].
5.14 Lenna Let
(I) 0-- A' -- A  A" ---0
be  exact aequence of K-modulea, md let B be a K-moVie. n t induced
eence
(2) A'  B A  B A   BO
(whose homomorphi are t homorph of (1) teored with the
identity momorphism of B) is exact, but A'  B  A  B ia not nessarily
injective. If, wer, either A # or B ia torsionless, then the full sequence
($) O A'  B A  B A #  B 0
exact.

Cochain Complexes 173
OVc should point out that (3) is the first place in this chapter where the
fact that the sin S K is actually a principal ideal domain is used.)
$.15 Theorem Let
(1) 0-.- 8'--- 8--- 8"-- 0
be an exact sequence of sheaves over M, and let 5 r be also a sheaf over M.
Then if either 5 r or gw is torslonless, then the sequence
(2) 0-- 8' ® .'--, g ® .-- 8" ® .'--, 0
If, in addition, either 5 r or 8' is a fine heaf, then the equence
0 -- P(S' ® ') -- P(S ® ') -- P(S W ® ') -- 0
x, gooF The exactness of (2) follows from 5.14. If either 5" or 8' is
fine, then, according to 5.10, $' ® 5 r is fine; and this together with
(2) above, 5.11(2), and 5.12 proves that (3) is exact.
COCHAIN COMPLEXES
5.16 DefintUons A cochain complex C* consists of a sequence of K-
modules and homomorphisms
(I) "'" -- C *- "* C* --* C *+ -*""
defined for all integers q such that at each stage the image of a given homo-
morphism is contained in the kernel of the next. The homomorphism
C' -- C '+x (which we shall refer to as do, or simply d if the index q is not
needed for clarification) is called the qth coboundary operator. The kernel
Zo(C *) of d o is the module of qth degree cocycles of the cochain complex
C*, and the image Bo(C *) of do-x is the module ofqth degree coboundarie$.
The qth cohomology module Ho(C *) is defined to be the quotient module
(2) Ho(C *) : Zo(C*)/B(C*).
Let C* and D* be cochain complexes. A cochain map C* -- D* consists
of a collection of homomorphisms C o -- D o such that for each q, the diagram
(3) T T
C o  D o
commutes. It follows from (3) that a cochain map sends the module of
q-cocyles of C* into the module ofq-cocycles of D* and maps the module of
q-coboundaries of C* into the module of q-coboundaries of D*, and thus
induces a homomorphism of the cohomology modules
(4) Ho(C *) -- Ho(D*).

174 Sheaves, Cohomoloy, and the de  heorem
The compos/tion C* --,. E* of two cochain maps C* --* D* and D* --*
induces on the cohomolosy modules the homomorphism H.(C*) --. H.(£*),
which is the compos/tion of H.(C') --* H'(D') and H'(D') --*
A sequence of cochain maps
(5) O----C*---- D*----E*----O
forms a short exact seence if for each q,
(6) 0---- C' ----4.
is a short exact sequence of K-modules. A homomorphim between short
exact sequences 0 -- C* -- D* -- E* -- 0 and 0 -,- * -- * -- * -- 0 of
cochatn complexes consists of cochain maps C* -- C*, D* --/*, and
E* -- * such that we have a commutative diagram:
O-- C* - D* -- £* -- O
(7)  
0C*/* *0.
5.17 Proposition Given the short exact sequence 5.16(5) of cochain
maps, there are homomorphixm
(I) H'(E*)  H'+I(C *)
for each q uch that the sequence
(2)
• "" -- H'-I(E *)  H'(C*) -- H'(D*) -- HI(E *)  H+I(C *) --...
i exact, and uch that given the homomorphin 5.16(7) of short exact equence$
of cochain complexes, the diagram
H'( E*) 
H'(£*) 
commutes.
oov To define , we consider the commutative diagram (4), where
for convenience we have labeled particular homomorphisms. Let ¢
be a cocycle in E,. Since  is surjective, there is an element   D"
such that ()= ¢. Since the square () is commutative, and since
¢ is a cocycle, it follows that d() maps to zero in £,+1; therefore,

Cochain Complexes 175
(4)
by the exactness of the horizontal sequences, d(O) must be in the image
of/3. Thus there is an element ' ¢ C +1 such that /3(o') = d(O). It
follows from the commutativity of the square ), the injectivity of
7, and the fact that d o d = 0 that e' is a cocycle. Now, e' is. not
uniquely determined by e because of the choice involved in the selection
of  e D such that =() --- e. However, it is easily seen by a quick
chase around the square (]) that e' is uniquely determined by e up to a
coboundary. Thus we have a well-defined map
() Zo(E*) ..-,,
which is easily seen to be a homomorphism. It is also readily seen that
B(E *) is in the kernel of (5); hence (5) yields a homomorphism defined
on the quotient Z(E*)/B(E*). This by definition is the desired homo-
morphism (1).
Checking the exactness of (2) involves checking exactness at H(C*),
at H(D*), and at H(E*)--and at each of these three stages we have
to check two inclusion relations. These six steps in the proof of the exact-
ness of (2) can be verified by simple chases around the diagram (4).
We leave the details to the reader.

176 Sheaves, Cohomology, and the de Rham Theorem
To prove (3), we have to follow some elements around the above
commutative lattice.
We start with a cocycle ¢ ¢ £' and lift it back to 8 ¢ D% which we
map to d(#)¢ D x, which we have seen is the imag_c of an element
¢' ¢ 0 +x, which in turn we map to " ¢ C '+x. Then G is a cocycle in
?.+x and is a representative of the cohomology class in Ho+x(C *) into
which the cohomology class of ¢ in H(£*) is mapped under the composi-
tion H(£ )  H'+x(C ) --* H'+x(C). On the other hand. we can
first map ¢ to  ¢ ', then rift back to  ¢ , map to d()
and lift back to 8' ¢ C'+L Then 8' is a cocycle in '+t which represents
the cohomology class in H'+X( *) into which the cohomology class of
¢ is mapped under the composition H,(E*) --, H'(*)  H'+X(*).
To prove (3), we need only check that o  -- 8' is a coboundary. Now
8 ¢ D' maps to an element  /, and elements  and  cf/ both
map to 8  J'; hence there is an element 7'  ' which maps to
One can easily check that dy =  -- 8', which completes the proof of
Proposition 5.17.
ARer a few trials the reader should find that these proofs by "diagram
chase" become quite routine, albeit tedious. In the future we shall leave all
such proofs as exercises.
AXIOMATIC SHEAF COHOMOLOGY
5.18 Definition A sheaf cohomology theory  for M with coefficients
in sheaves of K-modules over M consists of
(1) a K-module/P(M,$) for each sheaf $ and for each integer q,
(II) a homomorphism /P(M,$)--*/P(M,$') for each homomorphism
g --* g" and for each integer q, and

Axiomatic Sheaf Cohomology 177
(IH) a homomorphism /P(M,8")-*/P+I(M,8,) for each short exact
sequence 0 --* 8' --* 8 --* 8" --* 0 and for each integer q,
such that the properties (a)--(f) hold:
(a) IP(M,8) -- 0 for q < 0, and there is an isomorphism He(M,8)_
P(8) such that for each homomorphism 8 --* 8" the diagram
commutes.
(b) /P(M,8) = 0 for all q > 0 if 8 is a fine sheaf.
(c) If0--* 8'--* 8--* 8"--*0 is exact, then the following is exact:
• " -- Ho(M,S') -. H'(M,S) -- H'(,S') -- HI(,S ') --. • •
(d) The identity homomorphism id: 8--* 8 induces the identity
homomorphism id:/P(M,8)--* H'(M,8).
(e) If the diagram
8 8'
commutes, then for each q so does the diagram
H*(M,S) , H'(M,S')
(f) For each homomorphism of short exact sequences of sheaves
0---- 8' ---- 8 ---- 8" ---- 0
0 ---- ." ---- ." ---- " ---- O,
the oHowing diam mmutes:
(M,8")  HX(M,8 ")
w(n  3
The module H*(M,8) is lled e h cohomology module of M with
coefficients in the sheaf 8 lative to e homology ey .

178 Sheavex, Cohomology, and the de Rham Theorem
5.19 Definition An exact sheaf sequence
(1) 0 --- ' --- 'o --- 'i --- ', --""""
is called a. resolution of the sheaf ,'. The resolution (I) is called .ne
(respectively toraionleaa) if each of the sheaves ' is fine (respectively
torsionless).
With each resolution (1) of,' and each sheaf 8 we associate a cochain
complex
(2) -.. --- 0--, I'(0 ® s)--, I'(1 ® s) --, I'(l ® s) -*...
which we shall denote by P(* ® 8). For q  0 the module of q-cochalns
is P(, ® 8), whereas for  < 0 the module of q-cochains is the zero
module. Note carefully that this cochain complex does not contain the
module P(m¢' ® 8). The homomorphisms in (2) are those induced by the
homomorphisms  ® 8 - +x ® 8 which are the tensor product of the
homomorphisms of (I) with id: 8-- 8. The exactness of (I) implies that in
(3) ...--, 0--, e0 ® s- s' ® s--, el ® 8-,...
the image of each homomorphism is contained in the kernel of the next,
and this in turn implies that (2) is indeed a cochaln complex.
A homomorphism 8 -- 8' when tensored with the identity homomorph-
isms of the sheaves ' yields homomorphisms ' ® 8 -,- ' ® 8'. These,
in turn, induce homomorphisms P(' ® 8) -,- I'( ® 8'), which commute
with the coboundary homomorphisms of the respective cochain complexes
and thus determine a cochain map
(4) P('* ® 8)  P('* ® 83.
5.20 Existence of C0homology Theories We shall now show that
each fine torsionless resolution of the constant sheaf  z M × K,
(D 0---  --- '0 --- ' --- ', --- • • •,
canonically determines a cohomology theory for M with coefficients in sheaves
of K-modules over M. The existence of such resolutions will be amply
demonstrated in the later sections in which the classical cohomology theories
are discussed. In particular, see 5.26(7) and (I I). We therefore assume that
we are given a fine torsionless resolution (I). Then we obtain a cohomology
theory as follows:
(I) With a sheaf 8 and each integer  we associaXe the qth cohomology
module of the cochain complex I'('* ® 8); that is, we set
H(M,8) = H(P( * ® 8)).
(II) With each homomorphism 8-,- 8" and each integer q we associate
the homomorphism H*(M,8)--IP(M,8") induced, according to
5.16(4), by the cochain map P(* ® 8)-- P(* ® 8') of 5.19(4).

xiommic heaf Cohomolo ! 79
(HI) Each short exact sheaf sequence
0-- 8'-- 8 --* 8"-- 0
induces, in view of Theorem 5.15 and the fact that the  are fine
torsionless sheaves, a short exact sequence of cochain maps
0 - r(* ® S') --- r(* ® S)  r(, ® S') --- 0
with which, according to 5.1"/(I), there is associated a homomorphism
Hq(r( '* ® $"))-,-Hq+l(r( '* ® $')). This, by definition, is the
homomorphism H*(M,g")--, H*+(M,g') that we associate with the
short exact sequence 0 -,- g" --* g --, ;" --* 0 and the integer q.
The fact that axiom 5.18(d) for a cohomology theory is satisfied is immedi-
ately apparent. The remark immediately following 5.16(4) implies that
axiom (e) is satisfied. Axioms (c) and (f) are consequences of 5.17.
Now let ., be the kernel of f, --* f+l. Then it follows from the exact-
ness of (!) that
(2) 0 - r0 - eo - r, - 0
is exact; and since . is a subsheaf of the torsionless sheaf , then . is
also torsionless. It follows from Theorem 5.15 that for any sheaf  the
sequence
(3) 0-0 ® s- e0 ® s- . ® s-0
is exact; and from (3) and 5.11 (2) we obtain the exact sequence
(4) 0 - r(.0 ® s) - r(o ® s) - r(.. ® s).
In particular, P(. ® 8) -- P(q ® $) is an injection, so since the homo-
morphism
is the composition
(6) r(0 ® S)-- r(.,+ 1 ® s) -- r(,+l ®
it follows from (4) that the kernel of (5) is simply the submodule r(. ® $)
of p( ® $). Thus for q  0,
Now, if $ happens to be a fine sheaf, .then according to Theorem
the full sequence
(8) 0 -- r(r_ ® S) -- r(0_ ® S) -- r(.0 ® s) -- 0
is exact; and (8) together with (7) implies that
H(M,)---0 forq0.
Thus axiom (b) is satisfied.

180 Sheaves, Cohomologv, and the de Rham Theorem
Finally, it is evident that /(M,$) .= 0 for f < O. and for f .= 0 the
isomorphism
(9) _. r.  .
induces, for an arbitrary sheaf 8, an isomorphism
and thus an isomorphism
whach clearly satiti
r() _
r(') _
for each homomorphism  -* '. Thns aom (a) i$ aticd.
5.21 Defln/tlon Let ., and J be two sheaf cohomology theories on M
with coefficients in sheaves of K-modules over M. A homomorptgsm of the
cohomology theory  to the theory . consists of a homomorphism
(I) H'(M,S) //0(M,S)
for each sheaf $ and for each integer q, such that the following conditions
(a) For q = 0, the diagram
(b)
hold:
(c)
H°(M,S) 
For each homomorphism 8 -*" and each integer q the diagram
H'(M,S) -
//O(M,) -//°(M")
commllteso
For h sho exact suen of sbeav
00
the fllowing diagram commums for ch inmg q:
w(  (,)
,(  ,+(,).

,4xiomatic Sheaf Cohomoloy |81
An iomorphism -  is a homomorphism in which each of the
homomorphisms 1P(M,8)-II'(M,$) are isomorphisms. The Identity
homomorphim of. with itself is the liomomorphism . - . which assigns
the identity homomorphism 1P(M,8)- H(M,8) to each sheaf 8 and each
integer q.
We are now going to prove that there is a unique homomorphism
between any two sheaf cohomology .theories on M. However, first we need
to introduce the notion of the sheaf of germs of discontinuous sections of a
sheaf g.
5.22 The Sheaf of Germs of Discontinuous Sections of g Let g be
a sheaf on M. By a dscontinuous section of g over the open set U  M we
mean any map f: U--* $, continuous or not, such that r ofz id. The
assignment to each open set U  M of the module of all discontinuous
sections of g over U yields a presheaf whose associated sheaf g0 is called the
sheaf of germs of dscontinuous sections of $.
The important property of $0 that we shall need is that $0 is always a
fine sheaf. For let {U} be a locally finite open cover of M. Choose a refine-
ment {V} such that   U for each  (Exercise 3, Chapter I). (For such
a refinement on a general paracompact Hansdorff space, see [13, Ch. 5,
Problem V(a)].) Associate with each point of M a set V containing it, and
then for each i define a function q on M to have the value I at all points
associated with V and to have the value 0 elsewhere. It follows that
supp %  U and  q -- I. We associate with ? an endomorphism 
of the presheaf of discontinuous sections of $ by defining
(I) (s)(m) = ,(m(m)
for each discontinuous section s of $ over an open set U  M and each
m  U. The presheaf endomorphisms  induce endomorphisms l, of the
sheaf So of germs of discontinuous sections of $. It is immediate that the
{/} forms a panitlon of unity of go subordinate to the locally finite cover
{U,} of M. Thus go is a fine sheaf, as claimed.
There is a natural injection of g into go; namely, each element of the
stalk $, is the value at m of a (continuous) section s of $ defined on some
neighborhood of m, and we map s(m) to the germ ors at m in So. If we let
 denote the quotient sheaf go/g, then for each sheaf g we have constructed
a short exact sequence
(2) 0  S - S0 -  - 0
in which the middle sheaf is fine.
5.23 Theorem Let 3F and  be cohomology theories on M with coe:lcients
in sheaves of K-modules over M. Then there exists a unique homomorphism of
,,'f to.,.

182 Sheaves, Cohomoloy, and ¢he de Rham Theorem
PROOF We first prove uniqueness. Suppose that we have a homo-
morphism of . to ., and let a sheaf S be given. Then it follows
from the axioms 5.18 and 5.21 applied to the exact sequence 5.22(2),
in which, you recall, the sheaf So is fine, that we have a commutative
diagram
r(So) - r() - t(M,S) - 0
r(s,) -* A,(M,s) -. 0;
and for each q  2 we have a commutative diagram
0 - w-,(ta,) - W(M,S) - 0
(2)  
0 - f/,-(ta,) - F/,(M,S) - 0.
Note that in both (I) and (2) the rows are exact. Now uniqueness of the
homomorphisms 5.21(I) follows for q  0 from 5.21(a), follows for
q = I from (I), and follows inductively forq > I from (2).
We now show the existence of a homomorphism of the theory Jf
to the theory .. We define the homomorphisms for -- 0 by 5.21(a),
and it follows immediately that 5.21(b) is satisfied for the case  = 0.
We define the homomorphisms 5.21(I) for q  I by (I), and inductively
for  > I by (2). To prove that this indeed defines a homomorphism
of  to ., it remains to show that 5.21(b) holds for   0 and that
5.21(c) holds for all q.
Now, 5.21 (b) follows for  = I from the following lattice constructed
from two copies of (I):
In the lattice, the commutativity of the right-hand face (]) is forced
by the apriori commutativity of all the other faces, and 5.21(b) follows
inductively for q > 1 from the analogous lattice constructed from (2).

Axiomatic Sheaf Cohomology 183
As for 5.21 (c), suppose that a short exact sequence
(4) O,gO
is given. Let ,o and $0 be the sheaves of germs of discontinuous
tions of , and $ respectively. The composition ,  $  80 is an
injection_ of , into 80 ; let f be the quotient sheaf 80/,. As in 5.22,
let , = ,o],._ Then there are uniquely determined homomorphisms
• " -- ff and ,  ff such that the following diagram, in which the rows
are exact, commutes:
0----+,--.-+ $ ----'----0
(5) 0"--'-+ ,'--'-+ 80 ---* ff--- 0
0 ---- , ---- .o ---  ...._ O.
It follows from the cohomology axioms 5.18 applied to (5) that there
are commutative diagrams
• ..  r($)  r(-)  H(M,..)---- • • •
(6) ...  1"($,)  l"(f) --.-+ (,.) ._._+ 
• "" ---*" I"(D ---*. 1" ) ----* (M,) --.-. 0
and (for q > l)
• .. /P(,) .--.-./-/,(,.'._._./-/,÷(,)  ...
(7) 0  H'(M,f)  H,÷a(M,)  0
  m(,)  H,÷a(,)  .
It follos from (6) that the homomorphism H(M,, ") --/P(M,,) is
the composition
(8) H°(M,..q ") . F(.9 r) --. r(iimr(so), r(a)iimr(,o)  n(M,$),
and it follows from (7) that the homomorphism HO(M,.9 r) --. H+(M,.)
for q > 1 is the composition
(9) H(M,,9 r) -- m(M,f#) . It*(M,) - It,+,(M,).

184 Sheaves, Cohomolog, and the de Rham Theorem
From (8) and the corresponding sequence for the jo theory we obtain
the diagram
He(M,oq r') - r(s) --. r()/imr(So)  r()/imr(to) /(M,.)
(10) id id lid
in which the tint nd last quares commute by the definitions of the
homomorphisms H(M,) -- $/o(M,) and /P(M,t) --
•nd the middle two quares trivially commute. Thu 5.21(c) is proved
for q m 0. From (9) and the corresponding equence for the o' theory
we obtain the diagram
/o(,-) -.,/o(,) /o(,)  (,)
in which th lt squa commut by dnion of  homomohism
+(M,  +(M,), and tbe flt two squa ute by
5.21(b). Thus 5.21(c) follows for   I, and  proof of Thrum
5.23 is complte.
ro A omomorpism o the comolo try  w t¢ tory
must necessarily be an isomorphism. Consequently, any two
theories on M with coecients in sheas of K-moles or M are uely
isomorphic.
r Assume that a homomohism    is Oven. By eorem
5.23 there must also exist a homomohism   . e mi-
tion      must necsafily, by uniquens,  e identity
homomorphism of , d similarly the msition  
must  the identity homomohism of . It follows that the homo-
mohisms    and    are th imorphisms.
5.24 Assume that
is a commutative diagram in which the rows are fine torsionless resolutions
of the constant sheaf .r'. We have seen in 5.20 that each of these resolutions
determines a cohomology theory. Denote the cohomology theory obtained
from the top resolution by Jf and that obtained from the second resolution
by .. We claim that the "homomorphism" (1) between these two fine
torsionless resolutions canonically induces a homomorphism, and therefore,
||

Axiomatic Sheaf Cohomology !85
by the .corollary of 5.23, induces an isomorphism, 3f'--. o@. For each
sheaf 8, (1) induces a cochain map I'('* ® 8) --, I'(* ® 8) from which we
obtain a homomorphism tP(M,8) --, lq(M,8) for each integer q. That the
axioms 5.21(a), (b), and (c) for a homomorphism of cohomology theories
are satisfied is easily demonstrated and will be left to the reader as an
exercise. This procedure for obtaining a homomorphism of cohomology
theories will be used in 5.36 to obtain an explicit isomorphism of the
de Rham and differentiable singular cohomology theories for a differentiable
manifold M.
5.25 Theorem Assume that 3f  is a cohomology theory for M with
coefficients in sheaves of K-modules over M. Let
O) o So,...
be a fine reso/ution of the sheaf 8. Then there are canonical isomorphisms
(2) n  (u,s) for all q.
pltool From the exactness of (1) it follows that 0 - P(8) - P(We) "
P(W2) is exact, whence He(g,$)-- - P(8)--- He(P(W*)). Now let
.r'q be the kernel of W, - W,+2 for q  1. Then the exactness of (1)
implies that
(3) 0  ro  o  r,+  0
is exact for q  1, and that
(4) 0-.- $ - e -- .r' -- 0
is exact. It follows from the long exact cohomology sequence for (4)
and from the fact that We is a fine sheaf that
(5) I(M,$) --- H-(M,.") for q > 1
and
(6) /P(M,8)
By repeated application of the long exact sequence associated with (3)
and of the fact that the sheaves ' are fine, we obtain for q > 1
(7) H*(M,8)
-- r(,)/Im r(t',)  ,(r(t',)).
This completes our development of axiomatic sheaf cohomology except
for the multiplicative structure, which is discussed in 5.42. We shall now
consider four sheaf cohomology theories for a differentiable manifold M--the
Aiexander-Spanier, de Rham, singular, and ?ech. In view of the corollary
of 5.23, these theories will be uniquely isomorphic. We shall also define the
classical versions of these theories, and prove that they are canonically
isomorphic with sheaf cohomology with coefficients in constant sheaves.

186 Sheaves, Cohomoiogy, amt the de Rham Theorem
THE CLASSICAL COHOMOLOGY THEORIES
Aiexander-Slmnier Colmmolofy
5.26 For the Alexander-Spanier theory, M need only be a paracompact
Hausdorfl' space. Let U = M be open, and as usual let/f be a fixed principal
ideal domain. We use U "+1 to denote the (p + 1) fold cartesian product
of U with itself. Let A(U,K) denote the/f-module of functions U
under pointwise addition. For each p  0 we define a homomorphism
(1) d:
by setting
(2) df(me ..... m,+) = , (--l)f(me, .... m" ..... m+)
for each f A'(U,IO and (me .... , m,+)  U +l, where --- over an entry
means that that entry is to be omitted. It is a straightforward exercise to
check that d o d ffi 0, so for each U = M we have a cochain complex
d d
(3) ...  0 --. ,°(V,  ,I(V, d. ,l(V,  .-- . . .
which we shall denote by A*(U,K), and in which the modules of q-cochains
for q
and there is a corresponding restriction homomorphism
(4) pv.v:
The collection
(5)
forms a presheaf of/f-moduls onM called the presheaf of Alexander-
Spanter p-cochains. Observe that these presheaves satisfy 5.7(C,), but for
p  1 do not satisfy 5.7(C).
The associated sheaf of germs of Alexander-Spanier p-cochains we denote
by "(M,K). The homomorphisms d give presheaf homomorphisms
{A'(U,K);pv.. }--* {A+(U,K);pv..} for each p  0, and induce sheaf
homomorphisms (which we shall denote by the same symbol d):
(6) d: /'(M,K) --- /*+' (M,K) (p  0).
These homomorphisms, together with the natural injection of the constant
sheaf  --./°(M,K) which sends k  .¢', to the germ of the constant
function k at m, give us a sequence
d d d
(7) 0 --  -- ./O(M,K)  ./*(MJ0  ./*(M,K) --- • • " .

Alexander.$panier Cohomology 187
We claim that (7) is a fine torionless solution of the constant sheaf
f'. The elements of I'(M,K) are equivalence classes of functions with
values in K, and K is an integral domain; hence .a'(M,K) is torsionless.
To prove that .'(M,K) is a fine sheaf, let {U} be a locally finite open cover
of M, and take (as in 5.22) a partition of unity {?} subordinate to the cover
{U} in which the functions qh take values 0 or 1 only. For each i we define
an endomorphism  of A(U,K) by setting
($) (f)(ms,. • •, m,) -- ¢,(ms)f(ms .... ,
The endomorphisms 't commute with restrictions and thus determine
presheaf endomorphisms of {A'(U,K): per.v}. Let !:
be the sheaf endomorphism associated with . Then it follows readily that
supp !  U and that  it m 1. Thus the sheaves ./'(M,K) are fine. The
exactness of (7) follows directly from the fact that we have exactness on the
presheaf level. That is, if U  M is open, then
(9) 0 .. K- Ao(U, IO d. Ax(U, IO d. d
At(U,K) --- • • •
is exact. Here K is injected into A°(U,K) by sending k  K to the function
that has the constant value k on U. That (9) is exact is apparent at K and at
Ao(U,K), and follows elsewhere from the fact that d o d m 0 and that if
f  A(U,K) with p  1 and df m O, then f  dg, where g  A-x(U,K) is
defined by
(10) g(m. ..... m,-O - f (m, too. ....
for some arbitrary but fixed m  U. Thus (7) is a fine torsionless resolution
of f'.
The explicit exhibition of the fine torsionless resolution (7) of the constant
sheaf J" completes the proof in 5.20 of the existence of a cohomology
theory for M with coefficients in sheaves of K-modules. Thus by setting
(as in 5.20(1))
(l l) ® s)),
we obtain a cohomology theory with which, according to the corollary of
5.23, any other sheaf cohomology theory for M with coefficients in sheaves
of K-modules is uniquely isomorphic.
We shall now define the classical Alexander-Spanier cohomology modules
of M with coefficients in a K-module G, and show that they are canonically
isomorphic with the cohomology modules H*(M,) with coefficients in the
constant sheaf f  M × G. We let A(U,G) denote the K-module of
functions U +1 -, G, and similarly replace K by G and J" by f in the con-
structions (1) through (7). Let
(12) Ag(M,G) - {f  A(M,G): p,.M(f)  0 for all m  M}.

188 Sheaves, Cohomology, and the de Rham Theorem
Recall that p,,. is the homomorphism which assigns to each element
of P(M,G) its equivalence class in the stalk over m of the associated
sheaf .'(M,G). A(M,G) is a submodule of A'(M,G), and the extent
to which A(M,G) is not zero measures the extent to which the preshcaf
{A,(U,G);pv.v } fails to satisfy 5.7(Cx). The homomorphism (1) restricted
to Af(M,6) has range in A+x(M,6) and thus yields homomorphisms on
quotients
(13) A'( M,G)/ A( M,G) --,. A+x( M,G)/ A*( M,G).
The sequence of modules and homomorphisms given by (13) for p  0
form a cochain complex which we shall denote by A*(M,G)/A(M,G),
and in which the modules of q cochains for q < 0 are as usual all assumed
to be zero. The classical Aiexander-Spanier cohomology modules for M with
coecients in the K-module G are given by definition by
(14) H_s(M ;G) = He(A*(M,G)/A;(M,G) ).
Since the sequence (7) with K replaced by G and  by f is a fine resolu-
tion of f, it follows from 5.25 that there are canonical isomorphisms
(15)
That there are canonical isomorphisms
(16) H._s(M ;G) -. I'I(M,)
followsfrom (14) and (15) and from the fact that the natural cochain map
(17) A*(M,G)/Af(M,G)  I'(.t*(M,G) )
is an isomorphism. This in turn is an immediate consequence of the following
proposition.
5.27 Proposition Let {Sv ;P.v} be a presheaf on M satisfying 5.7(Ct),
and let g be the associated sheaf. As in 5.26(12), let
(1) ($)e m {s¢ $.u: p,.t(s) m 0 for all m
Then the sequence
(2) o--. (s),  $ 2. l'(g)  o
is exact.
toov Since y is.the homomorphism which sends s  $ to the global
section m -* p,,.(s) of 8, the exactness of the sequence (2) at
is the result of the definition of (S)e. Thus we need only prove that
7' is surjective. Let t  I'(8). Then there is a locally finite open cover
{U.} of M, and there are elements s,  Sv, such that
(3) (s.) - ] v..

de Rham Cohomology 189
Let {V.} be a refinement such that   U.. Let Im be the (finite)
collection of all those indices 0 for which m ¢ '. Choose a neighborhood
Wm of m such that
(a) W.Vm  if /¢
(b) w, ['iv,,
(c) Pw.,v.(s.)  pw..v..(s..) if 0, 0' ¢ I..
Let sm¢ 5-. be e mmon image of e elemen in (c). en for
1 n and m in M,
(4) p.,..(s,) - p....(s,).
For let p ¢ W.  W.. en it follows from (a) t
 let • ¢ I. en aording to (c),
( s.  p..v.(s and s. 
so at
(6) p....(s,) = p.,,v,(s,)= p....(s,),
wch prov (4). It follows from (4) d e proy 5.7() at the
is an element s ¢ $ such at
and it follow from (, (5), and (3) at (s)  t.
de Rham Cohomology
5.28 In this section we shall take the principal ideal domain K to be the
real number field 11. Let U  M be open. The set of differential p-forms
on U forms a real vector space, which we have denoted by E(U'). These
real vector spaces together with ordinary rmtriction homomorphisms
Pv.v form a presheaf
(t) {E'(o') ;pv.v}
which satisfies both 5.7(C1) and (C2) and thus is complete. From the exterior
derivative operator d we obtain prmheaf homomorphisms
d
(2) {(o');pv.,} --* {E'l(o');pv,v} (p > 0).
We shall denote the associated sheaf of germs of differential p-forms by
MS(M), and we shall retain the symbol d for the sheaf homomorphisms
induced b,y (2). The constant sheaf '  M ×  can be naturally injected
into O(M) by sending a  ',,, to the germ at m of the function with constant
value a. Thus we have a sequence

190 ,Sheaves, Cohomology, and ¢he de R/tam Theorem
d
(3) o -..  -.. 'o(u)  ,r,(u)  ,r,(u) --. • •..
We claim that (3) is a fine torsionless resolution of the constant sheaf .
Partitions of unity can be constructed for the sheaves ,f'(M) exactly as they
were constructed in 5.10 for '®(M) -- ¢fO(M). Thus the sheaves d"(M)
are all fine. They are certainly torsionless since they are sheaves of real
vector spaces. The sequence (3) is clearly exact at  and at ¢fe(M). Since
the exterior derivative operator satisfies d o d -- 0, the same is true of the
sheaf homomorphisms in (3); hence the image of each homomorphism is
contained in the kernel of the next. That the sequence is actually exact,
and therefore a resolution, is a consequence of the corollaries of the Poincar
lemma 4.18. We shall return to the resolution (3) in a moment. First,
we comment about the structure of the Poincar lemma.
$.29 Retnark The collection of operators At in the Poincar lemma
4.18 is an example of a very useful tool in the cohomology theory of cochain
complexes called a homotop), operator. We shall have further occasion to
make use of such operators. They arise generally in the following situation.
Suppose that we have two cochain maps, call themfand g, between cochain
complexes C* and D*, and suppose that we want to show thatfand g both
induce the same homomorphisms of the cohomology modules. This will
follow if we can find homotopy operators for f and g, namely, homo-
morphisms At: C t--* D t-I such that
ht+ o d + d o h " ft -- g
on C  (where d indicates the coboundary operators in both C* and D*).
For then if ¢t¢ C t is a cocycle, then f(ct) --gt(ct) up to a coboundary,
sof(ct) and gt(ct) lie in the same cohomology class. In the Poincar lemma,
we found homotopy operators (for k  1) between the identity cochain map
of the cochain complex
• .. -. o-. Eo(v) -+ E,(v) - ,(v) --...
and the zero cochain map, with U the open unit ball in Euclidean space.
Consequently, all of the cohomology groups of this cochain complex vanish
fork2 1.
5.30 Since 5.28(3) is exact, it is therefore a fine torsionless resolution of the
constant sheaf . According to 5.20, the resolution 5.28(3) gives rise to a
cohomology theory for M with coefficients in sheaves of real vector spaces
by setting
(1) (-) - (r(*() ® ))
for q  0 and for a r a sheaf of real vector spaces over M. In view of the
corollary of 5.23, this theory is uniquely isomorphic with the theory we
constructed in 5.26(11).

Singular Cohomology 191
If we take .5" to be the constant sheaf , then
(2) = w(r(¢.(u) ® w(r(¢.(m)).
Consider now the cohain complex F(d'*(M)):
(3) ..- -* 0 -* r('(m)-* r((m)-, rg,(m)-....
and the cochain complex E*(M):
(4) ... -. o-.
In view of the fact that the presheaves {E(U);pv.r,} are complete, it follows
from 5.8 (or is easily seen directly in this special case) that the natural
homomorphisms
(5) E(M)- F(¢'(M))
are isomorphisms. Since the homomorphisms (5) commute with the co-
boundaries of the respective cochain complexes (3) and (4), they induce a
cochain map E*(M)--* I'(d'*(M)) which is an isomorphism of cochain
complexes. Thus there are canonical isomorphisms
(6) w(r(¢.(M}))  W('(M)).
But tP(E*(M)) is the classical qth de Rham cohomology group H, x(M)
for M which we introduced in 4.13(1), namely, the quotient of the vector
space of closed q-forms (those annihilated by d) modulo the vector space of
exact q-forms (those in the image of d). Thus from (2) and (6) we obtain
canonical isomorphisms
(7) H'(M,gt) --- H]. a(M).
Singular Cohomolog),
5.31 In this section we again take K to be an arbitrary principal ideal
domain, and we let U = M be open. In 4.6 we introduced differentiable
singular simplices for the theory of integration on manifolds. Now we need
continuous singular simplices. We review the definitions. Recall that for
each integer p  1 we let
(1) A'---- (at ..... a,)': a,:l and each a,0.
--1
A" is called the standardp-simplex in '. Forp m 0 we set A e equal to the
1-point space {0}; and A e is the standard O-simplex. A (continuous) singular
p-simplex a in U is a continuous map a: A'-- U. Ifp  1, we define a
differentiable singular p-simplex in U to be a singular p-simplex a which can
be extended to be a differentiable (C ®) map of a neighborhood of A • in
 into U.

12 Sheave.v, Cohomolo, and the de Rham Theorem
We shall let $(U) denote the free abelian group generated by the singular
p-simplices in U. Elements of $,(U) are called singular p.chains with integer
coefficients. If a is a singular p-s/mplex in U, with p  I, then its boundar),
is defined as in 4.6(4) to be the singular (p -- I) chain
(2)  =  (-
where o  is the tth face of  (see 4.6(3)). The bounda operator extends to
a homomorphism of S(U) into S_I(U) for each p  1, and satisfies
(as in 4.6(7))
(3) i o  = 0.
Let $(U,/() denote the/(-module consisting of functionsfwhich assign
to each singularp-simplex in Uan element of/(. Such anfis clled a singular
p-cochain on U. Scalar multiplication and addition in the module $(U',K)
are defined by
(kf)(o) -
(4)
(/+ f)(e) = f(e) + f().
Each cochain in S'(U,/() canonically extends to a homomorphism of
into/(. In fact, this determines an isomorphism of $(U,/() with the
module of homomorphisms of $(U) into/(. For convenience we shall at
times regard elements of S(U,/() as such homomorphisms. If V = U,
we let
(5) ,v.v: s'(vJO-* s,(t',/()
be the homomorphism which assigns to each fe S'(U,IO its restriction to
singular p-simplices which lie in V. Then
(6) {s'(v,/();,r.v}
forms a presheaf on M called the presheaf of singular p-cochains. Observe
that these presheaves, for p >. I, satisfy 5.7(Ct) but not (CI), and that the
presheaf (Se(U,lO;pr.v) (which can be canonically identified with the
presheaf (Ae(U,lO;pr.;,) of 5.26 since each singular 0-simplex in U can be
identified with a point of U) is complete.
Forp  I, we let S-(U,IO denote the/(-module consisting of functions
f which assign to each differentiable singular p-simplex in U an element
of/(. Such an f is called a d/fferentiable singular p-cochain on U. With
restriction homomorphisms defined as in (5), we obtain the presheaf
($-(U,/();pv.;,) of differentiable singular p-cochains on M. Since the
developments of the differentiable and the continuous singular cohomology
theor/es are nearly word-for-word the same, we shall pr/mar/ly discuss
the continuous case. Just keep in mind that analogous constructions apply
if $-(U,IO, for p >. I, is substituted for S(U,K). When substantial
differences ar/se they will be discussed.

$insular Cohomoloy 193
A coboundary homomorhism
(7)
is dfined by setting
(S) df() --
forf SP(U,K) and for ¢ a inl (p + 1) simplex in U. It foows from
item (3)
(9) d o d - o.
Sin d mmu with tfion homomosms, d elds a psheaf
homomohism
00) {s'(v,;pv.}  (s(v,;pv.} ;
and at the e e, in view of (9), d makes
d d
(I I) "'" -- 0  (U,  (U,  (U ..-
into a hain mplex (whe the modules ofqhains are ro for q < 0)
which we deno by
We denote e associat sheaf of ges of singular hns by
$(M, (and by $(M, in the diffentiable ) and tain d to denote
the induced eaf homomoms
d
02) S'(M,K)  S(M,.
Obsee that $0(M,K) is simply e sheaf of ges of functions on M with
values in K. e constt sheaf  can  canonically injecd into $o(M,K)
by nding k  , to the ge at m of the funion on M with consent
vue k. us we have a quen
d d d
and an analogous sequence with $(M,K) repined b $(M, forp  1.
We claim that in both the continuous and the differentiable ses, the sequence
(13) is a fine torsionless solution ofthe constant sheaf . That the sheaves
$(M, (and $(M, for   1) are all tordoess is a conquen
of the ft that they are all eaves ofgs of n tys of functions with
values in K, and K is an integral domain. To see that the sheaves $(M,
(and $(M,K) forp  1) are all fine sheaves, let a locally finite on ver
{U} of M be given, and ke (as in 5.22) a pafion of unity {9} subordinate
to the cover {U} in which e functions 9 ke vues 0 or I only. For
each i, we define an endomohism  of S(U, by Setting
O4) 7,(D(o) ffi ,(o(0)V(o),
where if p
mmute with stfictions, and thus deteine preshe endomosms

194 Sheave, Cohomology, and he de Rham Theorem
of ($t'(UdO;pej,}. Let l: $'(MdO -- $'(M,K) be the sheafendomorphism
associated with . Then it follows readily that supp ! = U and that
l m 1. Thus the sheave, St'(M,K) (similarly $(M,K) for p > l)_are
fine sheaves. The exactness of (13) is apparent at at e and Se(M,K), and we
know that do d .= 0. For the exactness of (13), it remains to be shown that
at each stage $'(M,K) forp > I the image of the preceding homomorphism
contains the kernel of the next. This will follow if we can prove that for
"sufficiently small" open sets U, iff is a singular p-cochain on U such that
dr,= 0 and ifp  I, then there is a singular (p -- I) cochain g on U uch
that f-- dg. Here we shall make use of the fact that M is a manifold and is
therefore locally Euclidean. It suffices to assume that U is the open unit hall
in dtm .v. To prove that if df,= 0 then there is a g such that dg ,= f, we
need only find a homotopy operator (see 5.29)
(IS) ,: S,(U,K)- S'-(U,K) (, > 1)
such that
(16) d o h + h+ 1 o d " id.
We define h, as follows. Letfe St(U,K), where U is the open unit hall in
IR da -u, and let a be a singular (p -- 1) simplex in U. Define
(17) h,(]')(¢O-
where /;,() is the singular p-simplex in. U which maps the origin in A"
to the origin in U, and which is defined for (al ..... a,) # 0 by
(18) ,(o')(al ..... a,) = a, • o" ai a, ..... a, a, .
Gcometrically, (a) is the cone in U obtained by joining, radially to the
origin:
, extends to a homomorphism S,_t(U)-4-S.(U). It follows from (18)
and the definition 4.6(4) of the boundary operator  that
(19) id .=  o +x + , o 

Singular Cohomolog), 195
on $,(U) for p  1. Before checking (19) in general, try the case in which
p ,ffi 1, where it turns out that
(20)
and where the corresponding picture is
(o, 1)  *
I (,o)
Now, (16) follows from (19) and (17) and from the definition (8) of the co-
boundary operator d. However, here is one point where the treatment of
the continuous case does not quite sutice for the differentiable case. For
if ¢ is a differentiable singular (p -- 1) simplex with p : 2, and if
is defined as in (18), then (¢) will be a continuous but generally not a
differentiable singular p-simplex--there are differentiability problems
at the origin. For example, if ¢ were a differentiable singular l-simplex in
U, then for/(¢) as defined in (18) to extend to be a smooth mapping on a
neighborhood of the origin in 2, one would at least have to have the image
of ¢ contained in a 2-dimensional plane passing through the origin in U.
This, of course, may not be the case. The defect is easily remedied by means
of a "smoothing" function. Let q denote the real-valued C ® function on the
real line defined by 1.10(3). Then q(t) takes values between 0 and 1, and has
the value 1 for t:l and the value0for t0. Then ifp:2, and if
is a differentiable (p -- 1) simplex, we define ,(¢) by
(21) ,(¢Q(az ..... a,) ffi q a • ¢ a2 a ..... a, a ,
 --1
whe, as in (18), we assume that ,(a) maps the origin in A" to the orion
in U. Now extend  to  differentiable on all of '- so that  and each
of its derivatives is bounded. en ,(o)is dned on all of ', providing
that we ag that ,(a) maps any point whe  a, ffi 0 to the ofin in U.
Moreover, ,() is differentiable of class C  on all of ,. e only ints
where problems arise a those for which  a ffi 0; d since (t) and aH
of its derivatives vanish faster than any lynomial in t as t  0, and since

196 Sheaves, Cohmnolo, and the de Rham Theorem
¢ and each of its derivatives is bounded on II-z, it follows that all derivatives
of],(¢) of all orders exist, and are continuous, and are zero at points where
 a ,= 0. This completes the probf that (13) is a fine torsionless resolution
of the constant sheaf Jr, both for the continuous and for the differentiable
singular theories.
Thus both resolutions (13), for the continuous and the differentiable cases,
give rise, as in 5.20, to cohomology theories if we set
H*(M,S) = H°(I'(S*(M,K) @
(22)
H*(M.S) = H°(I'(S*.(M,K) @ S))
for each integer q and for $ any sheaf of K-modules over M. In view of the
corollary of 5.23, these theories are uniquely isomorphic and, moreover, are
uniquely isomorphic with the theories 5.26(I I) and 5.30(I).
5.$2 Let G be a K-module. We let S'(U,O -') denote the K-module consisting
of functions which assign to each singular p-simplex in U an element of G.
Similarly, we me.y replace K by G and Jf by the constant sheaf  in the
constructions 5.31(4)-(13).
The claasical singular cohomolog), groups of M with coefficients in a K-
module G are defined in the continuous and differentiable cases by:
=
(1)
We shall now show that the classical cohomology groups (actually
K-modules) are canonically isomorphic with the sheaf cohomology modules
H*(M, gf). It follows from Proposition 5.27 that
(2) 0 --* So*(a,) --* S*(a,) --, r (s*(a,))--, 0
is a short exact sequence of cochain complexes (with a similar sequence in
the differentiable case). Thus if we prove that
(3) H°($(M,G)) ,ffi 0 for all q
(also in the differentiable case), then it follows from the long exact sequence
5.17(2) that there are canonical isomorphisms "
(4) /
Thus it follows from (1) and (4), and from 5.25 (applied to the fine resolutions
of @ obtained by replacing K by G and Jt r by @ in 5.31(13)) that we have
canonical isomorphisms
(5)

• ingular Cohomoloy 197
The proof of (3) for the continuous singular theory goes as follows.
(The differentiable case is identical.) It is trivially satisfied for q < 0 since
in this range the modules of the cochain complex S(M,G) are all zero.
The module 8o(M,O ) is also zero since the presheaf {Se(U,O);pv.v} is
complete. Thus IP(S(M,G)) n O. It remains to prove (3) for'q  1. Let
H .= {U,} be an arbitrary open cover of M. Let $(M,G) be the cochain
complex consisting of modules 5(M,G) of singular cochains f, with values
in G, defined only on "H-small" singular p-simplices, that is, defined only
on those singular p-simplices whose ranges lie in elements of H. Each
element of S'(M,G) determines an element of S(M,G) by restriction to
H-small simplices. The restriction homomorphismsju: $'(M,G)-- $(M,G)
yield a surjective cochain map
(6) .u: s*(a,)-, s(a,).
The kernels of the homomorphismsjx form  cochain complex £ such that
(7) 0 -- K -- s*(a,) -- s(a,) -- 0
is an exact sequence of cochain complexes. The key ingredient in the proof
of (3) is the fact that the cochain mapjx induces isomorphisms in cohomology.
Let us assume this for the moment. It follows from thelong exact cohomology
sequence for (7) that
(8) H*(Kx)  0 for all q.
So now letq  1, and letfbe a cocycle in S(M,G); that is, dr,- O. Then,
by the definition of S(M,63, there is an open cover H of M consisting of
sufficiently small open sets so that feK. It follows from (8) that there
exists g /t -x  S--x(M,63 such that dg ,- f, which proves (3).
Now we return to the proof tlmt the cochain mapjx induces isomorphisms
of the cohomology. Since we will work with a fixed cover II of M, we drop
the subscript II from jx. To prove that j: S*(M,G')--* Sx(M,G) induces
isomorphisms of the cohomology modules, we shall construct a cochain
map
(9) k: S(a,) -- S*(a,)
such that
(10) j o k  id,
whence the cochain mapj must induce surjections ofthe cohomology modules,
and such that there exist homotoly operators h,: S'(M,6")--* S'-x(M,G)
for all p such that
(11) h.x o d + d o h, ,- id -- k, o j,,
whence k oj induces the identity on cohomology, which implies thatj must

198 Sheaves, Cottomolog),, and the de lam Theorem
induce injections. Hence it will follow from (10) and (11) that j induces
isomorpbisms on the cohomology modules. The definitions of h and k
require a few preliminary constructions. The details become a little technical,
but the general idea is this. Let f¢ 5(M,G). Thus f is defined only on
H-small singular p-simplices. We want to define k(f) to be an element of
SS'(M,G); that is, k(f) is to be defined on all singular p-simplices. We will
define an operation "subdivision" to break large singular p-simplices into
chains of smaller ones. By subdividing any singular p-simplex suffciently
many times, we will obtain a chain of H-small singular p-simplices to which
we can then applyf. So we will define k(f) on the large singular simplex to
be f on the subdivided ones. Simplices which are already H-small will not
be subdivided at all. The technical difficulty which arises will be due to
the fact that the number of times that a singular p-simplex a needs to be
subdivided in order to obtain a chain of H-small simplices depends on a.
 q  1. A linear p-simplex in A* is a singular p-simplex of the form
(12) (al ..... a,)-* 1- a re+alva+':'+
(0 -. e for p , 0), canonically determined by the ordered sequence of
points re, • • •, v, in &*. We shalldenote such alinear simplex by
The identity map of A* onto itself is a linear q-simplex in A* which we shall
denote simply by A'. The free abelian group generated by the linear p-
simplices in A* we shall denote by L,(A*). The barycenter of a linear p-
simplex a -- (re ..... v,) in A' is the point
1 1
(13) b,,-  . + "" + 
Given a linear p-simplex a -- (. ..... v,) in A' and a point  ¢ A', we
define the join va of v and a to be the linear (p + 1) simplex (v,
in AL The join operation extends by linearity to L,(A'). A direct calculation
shows that ifp  1, then
We define subdivision homomorphirans Sd: L,(A)-*L,(A ) by setting
Sd == id for p == 0 and by etting
(15) Sd(a) --
for a a linear p-simplex in A q with p  1, and extending lineady to L,(A).
We define homomorphisms R: L,(Aq)-*L.(A ) by etfing R == 0 for
p == 0 and by etting
(16) R(a) - b,(a - s0(a) - R(aa))
for a a linear p-simplex in A  with p  1, and extending linearly to L(A).
It follows from (13), (14), (15), and (16), using an elementary induction
argument, that

$inBular Cohomoloy 199
(17)
8oSd-- SdoS,
oR + Ro .= id -- Sd
on L,(A*) with p  1. (The collection of modules L,(AD and homomor-
phisms : L,(A*) --* L_(A*) form what is called a chain complex in contrast
with a cochain complex 5.16(1) in which the homomorphisms go the other
direction. The first formula in (17) says that subdivision Sd is a chain map of
this chain complex with itself. The second formula in (17) shows that R
is a homotopy operator for the chain maps id and Sd, from which it follows
that Sd induces the identity map on the homology groups (ker
of this chain complex.)
We define homomorphisms Sd:S,(U)--Sp(U) and R:S,(U)--*
S,(U) for p  0 by setting
for a a singular p-simplex in U, and then extending linearly to $,(U). It
follows easily that formulas (17) hold on S,(U) for p : 1.
Now let ¢ .= (ve .... , v,) be a linearp-simplex in A'. Then the diameter
of each simplex of Sd(cQ is at most p/(p + 1) times the diameter of ¢, for
the diameter of a linear simplex ¢ is the maximum distance between any
two of the vertices v. Of any two vertices in a simplex in Sd(cQ, at least
one must be of the form [1/(k + l)](v. + .-. + v,) for I  k  p. The
distance from this vertex to the other vertex is less than or equal to the
distance to some vs, and
I'
(V,o +'-- + v, D -
1 I ' [ P diam
k+l :-o  p+l
We are now ready to construct the maps h, and k,. Let a be a singular
p-simplex in M. The open cover o'-x(L r) of A" has a Lebesgue number 6
[27, p. 122]. It follows that for s large enough, the diameter of each simplex
in (Sd)*(A ') is less than 6 where (Sd)" is the s-fold composition of Sd with
itself. Thus each singular p-simplex in the chain (Sd)o(a) lies in an element
of the cover H. Now let s(a) be the smallest of those positive integers s : 0
for which each simplex of (Sd)o(a) lies in an element of H. Then we can define
homomorphisms k,: S(M,G)-- S(M,G) by setting k, == id for p  0,
and for p  1 by setting
(19)

200 $aeave, Cohomolog, and he de  Teorem
Note that Sd raised to a negative power is to be interpreted as the zero
homomorphism, and (Sd)e--id. We also define homomorphisms
h : SP(M,G) -- $-I(M,G) by setting h, ,= 0 for p  1 and
for p  2. That (10) holds is obvious, and (11) follows immediately from a
straightforward calculation. Finally, the fact that the homomorphisms
k, yield a cochain map (9) follows from (11). For from (11) we obtain
(21) d o k, °Jr '= k.x °j,+x ° d.
Butj is a cochain map. Thus d o k, °A "= k.l o d oj,. Sincej is surjective,
it follows that d o k, .= k.x o d. This completes the proof that j induces
isomorpbisms in cohomology.
ech Cohomology
5.$$ In the Alexander-Spanier, de Rham, and singular cases we obtained
• a sheaf cohomology theory by exhibiting explicit £ne torsionless resolutions
of a constant sheaf, and we then proved that there were canonical iso-
morphisms of the classical cohomolo, modules with the sheaf cohomology
modules. The ech theory, by contrast, arises from a direct construction
that does not involve finding a fine torsionless resolution of f'. We will
define modules/'(M,$) for $ a sheaf of K-modules over M, and show that
the axioms 5.18 for a cohomology theory are satisfied. We again direct your
attention to the comment in the introduction to this chapter that for much of
the chapter, M need not be a differentiable manifold. In particular, M in
this section need only be a paracompact Hausdorff space.
Let 1I .= {U.} be an open cover of M. A collection (Ue .... , U,) of
members of the cover such that Ue ""  U, ¢/  will be called a
q-simplex. If ¢ .= (Ue ..... U,) is a q-simplex, its support ]¢] is by definition
The zh face of a q-simplex
(U0, .... U_x, U+x .... , Uq). Let C'(1I,$), for q  O, be the K-module
consisting of functions which assign to each q-simplex a an element of
I'($,JoJ), and let C(1I,$) -- 0 for q < 0. Elements of C*(1I,$) are called
q-cochains. With a coboundary homomorpbism
(2) d:
defined by
(3) df(a) "-', (-- 1)'Pl,l.l,,if(o),

(ech Cohomology 201
one obtains a cochain complex C*(H,$) whose qth cohomology module,
which we denote by/*(H,$), is called the qth (ech cohomology module of
(M,H) with coecients in $. A homOmorphism $ -. $' induces by composi-
tion a cochain map C*(H,$) -* C*(H,$') and thus induces homomorphisms
(4)
for each q.
A cochainfbelongs to Ce(H,8)jf and only if f assigns to each open set
U.  H a section ors over U.. fis a O-cocycle. that is df-- O. if and only if
for every 1-simplex (U.o,U.t). Thusfis a 0-cocycle if and only iffdefines a
global section of $ over M. Thus
(6) /0(1i,$) ,ffi p($).
We now consider the effect of refining the cover II. If a cover  is a
refinement of the cover II, then there exists a map p: -- II such that
V_ p(V) for each
then we let p(a) denote the q-simplex (p(V0) .... , p(V,)) of the cover H.
Now, p induces a cochain map p: C*(II,$)--* C*(,$) ifwe set
(7)
for fe C*(H,$) and for ¢ a q-simplex of the cover . This cochain map
induces homomorphisms
(8) ,*: '(a,s)  '(,s)
of the cohomology modules. We claim that ifp and r are both refining maps
of $ into H, then p* -- %* for cach q. As usual, we prove this by finding
a homotopy operator. If o -- (Vo ..... V.I) is a (q -- 1) simplex of the
cover $, we let
(9) , ffi (,vj,..., (v,), (v,) .... , (v._l)).
We then define homomorphisms h: Cq(H,$)--* C-1($,$) by setting
00) h.(f)(o) ffi Z(-y.,,f().
From a straightforward calculation, one obtains
(11) h+ i o d + d o h -- %-  ,
from which it follows that
ment of H, which we shall denote by
homomorphisms/(H,$) ---*/*($,$). Since the set of coverings of M forms:..,.
a directed set under the relation
then the homomorphism /*(H,$)---*/(,$) is the composition of th'..:-
homomorphisms /(H,$) ---*/q(B,$) and /(B,$) ---*/*(,$)---then th

202 Sheaves, Cohomoloy, and he de Rham Theorem
collection of modules /*(1,$) and refinement homomorphisms forms a
diret system. Thus one can form the direct limit module
(12) //'(M,$)
(Construction is completely analogous to the construction of the module
$, from the modules St] in 5.6.) /*(M,$) is the qth ech cohomology
module for M with coecients in the sheaf of K-modules $. The classical
qth ech cohomo/ogy modu/e/*(M;G') of M with ¢oecients in a K-module O
is by definition/*(MJ'), where  is the constant sheaf M < O.
We shall now show that the cch cohomology modules (12) give a sheaf
cohomology theory in the sense of 5.18. It is apparent that/'(M,$) =, 0
for q < O, and it follows from (6) that /e(M,$) .= I'($); thus 5.18(a) is
satisfied. The homomorphisms (4) induced from a homomorph/sm $ -* $'
commute with the refinement homomorphisms (8) and thus induce homo-
morphisms
(13) /'(M,$) --/'(M,$').
It is immediate that these homomorphisms satisfy 5.18(d) and (e).
Consider now a fine sheaf $ and an integer q > 0. Since every cover of
M has a locally finite refinement, in order to prove that/'(M,$) ,ffi 0, it
suffices to prove that/'(H,$) ,ffi 0 for each locally finite open cover 1I. Let
{!,} be a partition of unity for $ subordinate to the locally finite open cover
[ ,ffi {U,) of M. We shall define homomorphisms h,: C([,$)  C-X([,$)
for eachp  1. Letf C'([,$), and let o ,ffi (Ue, .... Ut) be a (p -- 1)
simplex of the cover 1. Then l,o (f(U,, Ue,..., Ut)) has support
in U,  Ue ""  U_x, so we can extend l, ° (f(U,, Ue, .... U,_t))
to a continuous section of $ over Ue  • • •  U,_x by extending it to be zero
outside U.  Ue ""  U-x. We consider l, o (f(U,, Ue .... , U_t))
as this section over Ue  • • •  U,_x. Now define
(14) h,(f)(a)
It follows that
(15) d°hp+h+lod,ffiid for p : 1.
So iff is a q-coycle with q > 0, then there is a (q -- 1) cochain g, namely
g- h,(f), such that dg ,ffif. It follows that /*(1,$),ffi 0. Thus axiom
5.18(b) is satisfied.

,ech Cohomolo 203
A short exact sheaf sequence 0 - g' - g -- gn.. 0 induces exact
sequences
(16) 0 --, c,(u,s,)  c,(u,s) --, c,(u,s').
Let C'(H,$ ") be the image of" C(H,$) in C(H,$'). Then the short exact
sequences
yield a short exact sequence of" cochain complexes
A refining map/:   H induces a homomorphism of"short exact sequences
of" cochain complexes
0  C*(,S') --,. C*(,)  2"(,')  0
and thus by 5.17 induces a commutative diagram of"the associated cohomology
sequences
(20)
• ...-, Ro-I(,S'),(,S') - '(,S - Ro(,S')  -I(,S') -. • •.
On passing to the direct limit, we obtain a long exact sequence
-- o(,s)-- R,(,s') - o+(,s') --- -.-.
We shall now prove that the inclusion cochain map (*(H,$') --
on passage to the direct limit in cohomology induces isomorphisms
These isomorphisms, together with (21), will then yield a long exact sequence
() ... -- 7-(u,s )  o(u,s')
-- 7(,s)-- ,(,s') - ,+(,s')---. • •,

204 Sheaves, Cohomology, and the de Rham Theorem
which proves axiom 5.18(c). The quotient modules
(24)
O,(u) -
together with the induced coboundary homomorphisms form a cochain
complex *(tl) such that
(25)
• o --,- C*(u,$') -,. c* (u,s') -,. C*(u) -,. o
is exact. It follows from the long exact cohomology sequence for (25),
upon passage to the direct limit, that (22) will follow if we prove that the
direct limit of the modules H'((* (L)) is 0 for all q. So let t[ be a locally
finite cover of M. That the direct limit of the modules/P((*(tl)) is 0 will
certainiy follow if we prove that for an arbitrary f¢ C'(tI,g') there is a
refinement /=: 3 --* t[ such that /=,(f) ¢ O(3,g'). Choose a refinement
D = {O=} of }J = {U=} such that O= c U, for each ,,. For each p ¢ M
choose a neighborhood Vp such that
(a) Vp c O, for some ,,.
(b) If Vp r O. # f, then Vp = U..
(c) Vp lies in the intersection of the D r. containing p.-
(d) If ¢ is a q-simplex of the cover }J, and p ¢ Il (so "p = Il), then
pv,.i.if(u) is the image of a section of g over
It is possible to satisfy (d) since there are only finitely many q-simplices of the
cover H which contain p. Now let 3 be the cover {Vp}, and for every p
choose Op ¢D and Up ¢ H such that Vp = Op = Up. Thus we have a
refinement/: 3 .-4. ]. Now let ¢r = (Vp. ..... Vp.) be a q-simplex of the
cover 3. Then Vp. Op# , 0[q, so by (b), Vp.C Up,. Thus
V'p, = Up, r... r Up, - I/()1. Therefore
l=,(f)(=) = plol.,,¢o, lf ( U, .... ,
= p,.,.v.." pv...,,.,,f(u,.....,
hence by condition (d),/=,(f) ¢ '(3,$').
Finally, axiom 5.18(f) follows readily from the above construction of
(23) by making use of 5.17(3).
Thus the (ech cohomology satisfies the axioms 5.18 for a sheafcohomology
theory.

The de Rhmn Theorem 205
THE DE  THEOREM
$.$4 Convention Since according to the corollary of 5.23, any two
sheaf cohomology theories on M are uniquely isomorphic, we shall consider
them  identified via their unique isomorphisms, and shall henceforth use
I-P'(M,$) to denote thepth Cohomology module of M with coefficients in the
sheaf $. With this convention,
O) H'(M,S) = '(M,S);
and given any fine torsionless resolution
(2) 0--. --. Co---C---C,-----.,
we have
(s) (,s)  (v(* ® s)).
$.$$ In the preceding sections we have obtained canonical isomorphisms
O) '-s(;)  '(M;)  '®(M;)  '(M;)
of the classical Alexander-Spanier, singular, difl'erentiable singular, and
Cech cohomology modules of a difl'erentiable manifold M with coefficients
in a K-module G over a principal ideal domain K. We have seen that each
of these is canonically isomorphic with the sheaf cohomology module
H(M,) with coefficients in the constant sheaf ft. If we take K to be the
field of real numbers, we can add the de Rham cohomology group He X(M)
to the above list of isomorphisms. In particular, we have canonical iso-
morphisms
(2) .(M) -- '(M,)  '®(;).
We shall now prove, with the help of 5.24, that the explicit homomorphism
from the de Rham to the differentiable singular cohomology theory obtained
from integration of forms over differentiable singular simplices yields the
canonical isomorphisms (2).
We define homomorphisms
for each integer p  0 by setting
(4) k,()(o)
for each differentiable p-form o on  and differentiable singular p-siple
 in M. It is an immediate consequence of Stokes" theorem 4.'/ that the
homomorphisms k induce a cochain map
(5) k: E*(M)  S*(M,R).

206 Sheaves, Cohomology, and the de Rham Theorem
Let
(6) k,*: HL(M)  HI®(M;)
denote the induced homomorphism of the cdhomology modules
vector spaces), k* is called the de Rham homomorphism.
(ieal
$.6 The de Rham Theorem The de Rham homomorphism k* is
the canonical iomorphsm 5.35(2) for each integer p.
mtOOF The homomorphisms 5.35(3) can be defined for arbitrary open
sets in M, and yield preshcaf homomorphisms
which commute, by Stokes' theorem, with the coboundary homo-
morphisms 5.28(2) and 5.31(10). Thus the induced homomorphisms
of the associated sheaves form a commutative diagram:
0---   o(M)  (M)  (M) --*.-.
o S®(M,) SL(.M,) S'(M,) .-..
Consider now the following commutative diagram of cochain complexes
in which the rows, according to 5.0(5) and 5.2(2), are exact:
0 -- e*(r#) ---- '('*(r#)) -- 0
() ,
o s*.o(,) s(,) '(s*_(,))- 0.
The cochain map (I) induces the isomorphisms He s(M) _. H(M,.).
We proved in 5.2(4) that  induces the isomorphisms
H(M,). That ) induces isomorphisms on cohomology follows from
the corollary of 5.23 applied to the homomorphism of sheaf cohomology
theories induced according to 5.24 by the homomorphism (!) of fine
torsionless resolutions of ,. Thus from the uniqueness of the iso-
morphism between sheaf cohomology theories, ) induces the identity
isomorphism of/P'(M,.). It follows that k* is the canonical isomor-
phism 5.35(2) for each integer p.
$.$7 Earlier, in 4.17, we stated  slightly different version of the de Rham
theorem in terms of singular homology instead ofcohomology. We shall now
see that there is a natural isomorphism
() HX®(M;)- ®.(;)*
which composed with k* yields the isomorphism 4.17(1). (For the definition
of the real differentiable singular homology group ,H',(M;) and other
relevant notation, the reader should review section 4.16.) The map (1) is
defined as follows. If f represents a cohomology class in H®(M;),
II

Multiplicative Structure 207
thenfcan be considered as a linear function on the vector space .,$,(M;)
of real differentiable singular p-chains in M. In particular, f is a linear
function on the subspace of ®$p(M;IK) consisting of the p-cycles, and f
vanishes on thep-boundaries in ®Sp(M;IK) since dfm 0; hencefdetermines
a linear function on the homology group ®H(M;). Since a real differen-
tiable singular coboundary necessarily vanishes on all p-cycles, each coho-
mology class in H®(M;IK) determines a weB-defined element of ®H(M;)*
independent of the representative f chosen. This defines the map (1). We
leave it to the reader as an exercise to establish that (1) is actually an iso-
morphism. Clearly the composition of (1) with k* is exactly the homomor-
phism 4.17(1). Thus 4.17(1) is an isomorphism.
$.$8 Remark The singular cohomology groups H(M;) are topo-
logical invariants; that is, homeomorphic spaces have isomorphic real
singular cohomology (see Exercise 19). As a consequence of th/s and the
isomorphisms 5.35(I). and (2), the de Rham cohomology groups are also
topological invar/ants of a different/able manifold.
MULTIPLICATIVE STRUCTURE
In the case in which $ is a sheaf of K-algebras over M, we shall make the
direct sum of the cohomology modules /@'(M,$) into an associative algebra
over K. But first we need a few preliminary constructions.
$.$9 Definitions Let C* and 'C* be cochain complexes. Their tensor
product C* ® 'C* is the cochain complex whose rth module is the direct
sum
(1)  C, ® 'C,
and whose rth coboundary homomorphism is the direct sum
(2)  (d, ® '(id), + (--1)'(id)p ® 'd.).
If aZ'(C *) and Z*('C*), then a® Z+*(C* ® 'C*); whereas if
  '(C*) and •  Z*('C *) (or vice versa, if  f Z'(C*) and - .*('C*)),
then  ®   P+(C* ® °C*). Thus there is a well-defined homomorphism
(3) I-P'(C*) ® Ho('C *) - H-'(C * ® "C*).
5.40 Lemma The tensor product of two torsionless K-modules, for K a
principal ideal domain, is again a torsionless K-module.

208 Sheaves, Cohomology, and the de Rham Theorem
,xooF Let t # 0  K, and let K/t be the K-module whose elements
are {k/: k /, and in which addition and multiplication by K are
defined by
(1)
let A be a K-module. We define a sequence of homomorphisms
(2) .4 .-, K/ ® .4 --, K ® .4 .-, .4.
The first homomorphism is defined by a  (2/2) ® a, the second by
2 (,/) ® a,  2 , ® a,, and the third by
The second homomorphism obviously has an inverse, so is an isomor-
phism; and the third is an isomorphism according to 5.9(2). The
composition (2) sends a ¢ A to a ¢ A. By definition, .4 is torsionless
if and only if the composition (2) has kernel zero for each 2 # 0  K.
Thus A is torsionless if and only if
(3) 0 .-, .4 .-, (K/t) ® .4
is exact for each 2 # 0  K. Now let A and  be torsionless K-modules,
and let  # 0  K. Then (3) is exact, and so since  is torsionless it
follows from 5.14 that
(4) 0--,.4 ® Z .-, (/) 8.4 ®Z
is exact, whence A ®  is torsi0nless.
5.41 Definition The definitions of the direct sum $  " of sheaves
and the direct sum $  " - $"  ', of sheaf homomorphisms $ -- $'
and .'-,-." are obtained by replacing ® by  in 5.9(3)-(8), with the
deletion of 5.9(7). Observe that the direct sum of fine sheaves is a fine sheaf.
5.42 The MultipHcative Structure Consider a resolution
(1) 0  .X- e --,.  _  ._,.... "
By the tensor product of the resolution (1) with itself we mean the sequence
(2) o  % ® ¢o-,.(% ® ¢)  (¢ ® %) ...
• ..--.  % ®o_.....
in which the homomorphism ,' ® ' is the composition ,._
•  ® , ' ® ', and in which the homomorphism whose domain is
 ', ® '. is the direct sum
(3)  (,, ® 0% + (--|)'(id), ® o).

Multiplicative Structure 209
If (I) is a fine torsionless resolution of .X¢', we claim that (2) is also a fine
torsionless resolution of.X¢'. That the sheaves in (2) are fine is a confluence
of the fact that tensor products and direct sums of fine sheaves are also
fine sheaves. From 5.40 and the observation that direct sums of torionless
K-modules are torsionless, it follows that the sheaves in (2) are torionless.
That (2) is exact, therefore a resolution, is easily seen at  and at o ® o,
and is proved elsewhere by applying the Kunneth formula to the cochain
complexes obtained for each m  M from
(4) ." .0--..'o®'o('o®'('®'o-"""
by restricting to the stalks over m. The extensive algebra necesry for a
proof of the Kunneth formula, which expresses the cohomology of the tensor
product of cochain complexes in terms of the cohomology of the individual
complexes, would not be particulaxly illuminating for our purposes, so we
shall simply refer the interested reader to Spanier [28, Ch. 5, 4, THEOreM 2],
where the Kunneth formula is proved in detail.
Let g be a sheaf over M. From the resolution (2) we obtain as in 5.19(2)
a cochain complex which we shall denote by I'(" ® " ® g). If g is a
sheaf of algebras over M, then the natural homomorphism g ® g -- g
determined by the multiplicative structure of the stalks induces homo-
morphisms
which in turn induce a cozhain map
(6) ® g) ® ® $) ® ® g).
We have, cording to .39(3), a well-defined homomorphism
() t(,g) ® °(,g) - t(r( ° ® g) ® r( ° ® g))-
in the case in which $ is a sbe of algebras over M, the cochain map (6)
induces, together with (7), a homomorphism
® tzo(  ,g)
This will define the multiplizative structure in sheaf cohomoloy. But first
we need to show that the multiplication defined by ($) is independent of the
resolution (l) with which we ben. Consider another fine torsionless
resolution
By tensoring the resolution (1) with the resolution (9) (construction com-
pletely malogous to the tensor product of (l) with itself iven in (2) nd (3)),

210 Sheaves, CohomoloFv , and the de  Theorem
we obtain another fine torsionless resolution
where the last map is inclusion. This homomorphism of resolutions induces
a cochain map P('* ® $) --,. P('* ® * ® $) which, according to 5.24
and the corollary of 5.23, induces the identity map H'(M,$) -,- Ha(M,$).
By applying these constructions to various resolutions and then tensoring
resulting cochaln complexes, one obtains a commutative diagram of cochain
complexes and cochain maps
(II)
from which is induced the following commutative diagram on cohomology:
From (12) it follows that the multiplicative structure (8) is independent of
the choice of (I).
It follows from the construction of (8) and the associativity of tensor
products that the multiplicative structure induced on  H(M,$) by (8)
p
is associative. Thus (8) makes  H(M,$) into an associative algebra over K.
The homomorphisms 'p ® a-' a ® p defined by cp ® c a--
(--l)ca ® c a induce a homomorphism of the resolution (2) with itself in
the sense of 5.24. According to 5.24, this homomorphism of (2) induces a
homomorphism of cohomology theories which by the corollary of 5.23 must

Multiplicative Structure 211
be the identity isomorphism. But the induced homomorphism/P'+*(M,g) 
//'*(M,g) sends u. v into (--1)my • u if u ¢/P'(M,g) and v ¢ H'(M,g).
Thus the multiplicative structure of  H'(M,g) satisfies the anti-commuta-
tivity rlation '
(13) u" v = (--1)v • u (u /P'(M,$); v /P(M,$)).
$.45 The de Rham Cohomology Algebra Exterior multiplication of
differential forms induces, by mapping ¢ ® ,,  ¢ ^ ,,, a cochain map
0)
E*(M) ® E*(M)  E*(M)
which together with the natural homomorphisms
(2) HL (M) ® H. (M)  H'+'(E*(M) ® E*(M))
of 5.39(3) induces homomorphisms
(3) L(M)  L (M)  (,
wch make the & sum  H a( into an aiafi gvb ov
, 
Ts is ¢ clsi mtiplifiv¢ sctu  d¢  homolo.
Now,  Ha( so inherits an siafivv algebra suctu from
gvbra struu 5.42(8) in shvafhomology via v noni isomosm
We shall now prove that these two algebra structures on  Hes(M ) are
identical. For each open set U c M, exterior multiplication induces homo-
morphisms
(4) E,(U) ® E,(U)-. E,'(U)
which commute with appropriate restrictions and thus yield presheaf
homomorphisms
(5) {E'(V3;pu.v} ® {E'(V3;pu.v}-*
which in turn induce sheaf homomorphisms
(6) ¢'(M) @ e(M) --,- ¢'+(M).
The homomorphisms (6) induce, as is easily checked, a homomorphism (in
the sense of 5.24) of the tensor product of the resolution 5.28(3) with itself
into itself:
0 --,   ,f'(M) ® °(M)  (¢°(M) ® (M)) • ((M) ® °(M))-.-
0  .  (M)  O(M) ...

212 $/a;, Coomolo, and te de Rham Teorem
The homomorphism (7) induces a cochain map P(g*(M)® g*(M))--*
I(g*(M)) which, according to 5.24 and the corollary of 5.23, induces
the kientity map I(M,)-- H'(M,). Consider now the following
commutative diagram of c o chain complexes:
r(¢,(u)) ® r(¢,(u) ® r(¢,(u))
(s) 1'
E*(M) ® E*(M) '  E*(M)
The diagram (8), together with 5.39(3), induces the following commutative
diagram on cohomology:
The composition of homomorphisms in the top row of (9) gives peecisely
the multiplicative structure 5.42(8) in sheafcohomology; whereas the compo-
sition in the bottom row of(9) is precisely the classical multiplicative structure
(2), and the first and last vertical arrows are the canonical isomorphisms.
Thus the classical algebra structure on /, xx(M) is identical with that
induced from the algebra structure on sheaf cohomology.
5.44 The Singular Cohomology Algebra This section will be written
in terms of the continuous singular cohomology. All considerat/ous,
however, apply in exactJy the same manner to the differentiable singular
theory. Let f¢ Sit(M,K) be a singular p-cochain, and let g ¢ $(M,K) be
a singular q-cochain. We shall define a singular (p + q) cochain f.--g
called the cupproduct offandg. Let o be a singular (p + q) simplex in M.
Define
(1) (f--- g)(o)
=]'( o k.-x o v,+-* ..
-,+,-x o • o k+x)g(# ° k, "+'-x o o "+*-* o--. o kJ),
where ifq = 0, then the first factor on the right-hand side of (1) is f(#);
and ifp ffi 0, then the second factor on the right-hand side of (1) is g(#).
The k's are the mappings defined in 4.6(2). In other words, (1) says that we
start with o and take the top face q times to get ap-simplex to which we apply
f, and we start with  and take the 0-face p times to get a q-simplex to which
we apply g. The multiplication on the right-hand side of (1) takes place in
the principal ideal domain K. Associativity, namely,
(2) f--- (g --- h)  (f--- g) .-- h,

Multiilicative Structure 213
follows from (1) and from the identity
(3) ko
. "',++,k+'+'-I o. • . o k++l"+a o ko+- o... o ko •
which follows from rpeated applications of 4.6(5). The bilinear map
(f,g) -f --- g inducs a homomorphism
(4) S,(M,K) ® S'(M,K)
We claim that
(5) (af) --  + (--1)'f--- (a) = a(f- ).
We suggest that the rader check (5) first for the case in which f and g are
1-cochains and ( is a 3-simplex. In this case some simple pictures will aid in
following the computation. In general, let v be a singular (p + q + 1)
simplex. Then from (1) and 5.31(8), and from rpeated applications of
4.6(5), we obtain
a(f,..
l--0
4-@+1
= : (-1)y( o k, o k'-' .... o k+l) S( o k o k0 '-' o... o ko')
+  (--1)YO" o/-...._,.__"+" o "'" o k+l) gO" ° k'o" • • o k'o
= ° " " ° -, ° -, o," o °" " • °
k
o • • • o1¢ ..0',
+1
+ (--1)'0" o -,+.+,'+' o • "" o kO (--1)'gO" o k +"
tO
o''" o ko + o k/)
= (df...g)()+(--1)'(f...dg)().

214 Sheaves, Cohomology, and the de P.ham Theorem
It follows from (5) that the homomorphisms (4) determine a cochain map
(6) S*(M, IO ® S*(M, IO
The cochain map (6) together with the natural homomorphism$
(7) r.(;K) ® H(;K)-,- H,+'(S*(M,K) ® S*(O)
of 5.39(3) induces homomorphisms
which give the diret sum  H(M;/t') the structure of an associative algebra
over K. This is the classical multiplicative structure of singular chomology.
Now,  H'(M;/t') also inherits an associative algebra structure from the
algebra structure 5.42(8) on sheaf cohomology via the canonical isomorphism
- HY,(M;ff)  H'(M,Tt). The proof that these two algebra structures
on  H(M;ff) are identical is exactly the same as he proof of the corre-
sponding statement for the de Rham ¢ohomology as given in 5.43(4)
through (9), but with the replacement of f*(M) by 8*(M,ff) and E*(M)
by $*(M,K), and so on.
The results of 5.43 and 5.44 provide the following more complete version
of the de Rham theorem 5.36.
5.45
(1)
The de Rham Theorem 7he de Rham homomorphism
k*:  H.a(M) - . HX®(M,I,)
is an algebra isomorphism.
SUPPORTS
5.46 Definition A family of supports on M is a family • of closed
subsets of M satisfying:
(a) The union of any two members of • is again in O.
(b) Any closed subset of a member of • is also a member of .
(c) Each member of • has a neighborhood whose closure is in .
Let O be a family of supports on M. We recall that M is assumed to be
at least a paracompact Hausdorff space and where necessary is actually a
differentiable manifold. If 8 is a sheaf over M, we let I'.(8) be the set of
sections of $ whose supports are members of . It follows from condition
(a) that I'.(g) is a submodule of I'(g). It follows from (b) that a sheaf
homomorphism g --* g' induces a homomorphism I'.8) --* I'.(g').

• uppor$ 215
Theorem 5.12 has an exact analog with supports. One needs only to
make a slight modification in the proof by using condition (c) to help choose
a cover which will guarantee that the resulting section of 8 actually lies in
I'(8). The details are left to the reader as an exercise.
A cohomology theory J. for M with supvorts • and with coecients
in sheaves of K-modules over M is defined exactly as in 5.18, with the excep-
tions that P(8) is replaced by P(8) in 5.18(a) and the cohomology modules
are now denoted by l-P(M,). The entire development of ths chapter can
be carried through for cohomology with supports with very few modifica-
tions. We shall briefly sketch those modifications.
The proof of existence and uniqueness of a cohomology theory with
supports (which will make use of the support version of Theorem 5.12)
proceeds as in 5.18 through 5.25 with the exception that when given a fine
torsionless resolution 0 - . - ° o - ° 1 - ° s -- • •, we now define the
sheal" cohomology modules by setting
(1) .m(,s) = w(r.(* ® S)).
Let {S v ;lv.v} be a presheaf with associated sheaf 8. We let .$M be
the submodule of SM consisting of those elements mapping into Fe(8 ) under
the canonical homomorphism S -- F(8). It then follows from Proposition
5.27 and the definition of eS that 5.27 holds if we replace 5.27(2) by
(2) 0 (S  )0 "-" .Su "-" P.(8) --, 0.
The classical cohomology theories with supports are defined by:
.n'_s(M;6) = H°(.A*(M,6)/A0*(,)),
.n. R() = H°(.E*(),
(3) .n1(;) = Ho(.s*(,)),
-
In the (ech. theory, the support of a q-cochain f Co(II,8) is by definition
the union of the supports of the sectionsf(() as ( runs over the q-simplices
of the cover If. If we set .C°(H,8) equal to the module of -cochains with
supports in q>, then the development of the (ech theory proceeds as before and
yields the (ech cohomology modules .°(M,$) with supports.
The resultant cohomology theories will depend on the family of supports
chosen. Ordinary cohomology is the same as the special case of cohomology
with supports in which q> is taken to be the family of all closed sets.
The isomorphism theorems and the development of the multiplicative
structure all proceed for supports exactly as before. In particular, we have
the de Rham theorem with supports:
(4) k*:
is an algebra isomorphism.
! I !

216 heave, Cohomolo, and the de IOuvn Theorem
EXERCISES
1 Let 8 be a sheaf over M. Prove that the mapping m
(the "0-section") is continuous.
2 Prove that sections of sheaves are local homeomorphisms and there-
fore are open maps.
$ Prove that if two sections of a sheaf agree at one point, then they
agree on a neighborhood of that point. Conclude that the set of
points on which a section is not zero is a closed set.
4 A C ® function on M determines a cross section of the sheaf ofgerms
of C ® functions '®(M). The set of points in M on which a C ®
function is not zero is an open set; whereas, according to Exercise 3,
the set of points in M where a section of '®(M) is not zero is a
closed set. Can you reconcile these two facts? Consider examples.
5 Prove that sheaf mappings are local homeomorphisms, and therefore
are open maps.
6 Prove that if two sheaf mappings agree at a point then they agree
on a neighborhood of that .point. Conclude that the set of points
where a sheaf mapping is not zero is a dosed set.
7 Complete the details'of the construction in 5.4 of.quotient sheaves.
8 Complete the details of the proof begun in 5.6 that /((g)) is
canonically isomorphic with g.
• 9 Prove that there is a canonical isomorphism (g ® ,'),, __.
• 10 Prove that the tensor product of two complete presheaves need not
be a complete presbeaf.
• 11 Let q be a C ® function on M. Define a mapping'®(M)- '®(M)
by sending f,, -* q(m) • f,. Here, as usual, f, denotes the germ at
m of a C ® functionfdefined on a neighborhood of m. Prove that
this mapping is not in general continuous and therefore cannot be a
sheaf homomorphism. (Caution: Errors are often made at this
point in the construction of partitions of unity on sheaves--review
the correct procedure for the construction of a partition of unity on
'®(M) given in 5.10.)
12 Make the necessary modifications in the proof of Theorem 5.12
to prove that if • is a system of supports on M and g-" is a
surjective sheaf homomorphism with its kernel a fine sheaf, then
I'.(g) - I'.(') is a surjection.
13 Carry out the details in the proof of the exactness of 5.17(2).

Exercise$ 217
14 Complete the details of 5.24.
15 Prove that the presheaves 5.26(5) satisfy 5.7(C), but for p  1 do
not satisfy 5.7(Ca).
16 Give an example of an exact luence of modules
O- A --. B--* C--* O
and a module D such that
O-. A ® D -. B ® D -. C ® D -. O
is not exact.
17 Find an example of a fine sheaf which has a subsheaf which is not
fine.
18 Prove that if you tensor a long exact sequence of torsionless sheaves
with a sheaf 8, then the resulting long sequence is still exact.
19 Prove that a continuous map f: M-- N induces in a natural way
a homomorphism
f,: H(N;G) -.*/a(M;G)
such that if g: N-- X, then
(g *f), ---- f, * g,,
and such that
(id), = id.
Conclude that homeomorphic spaces have isomorphic singular
cohomology.
20 Prove that 5.37(1) is an isomorphism. Keep in mind that these
vector spaces are generally infinite dimensional.
21 Prove that if o and - are closed differential forms all of whose
periods are integer-valued (see 4.17), then o ^ - has integer periods.

Throughout this chapter, M will be a compact oriented Riemannian manifold
of dimension n unless otherwise indicated. We will see that the ordinary
Laplacian (-- 1)  2/x2 has a generalization to an operator A on differential
forms, known as the Laplace-Beltrami operator. Our main objective in this
¢hapter is a proof of the Hodge decomposition theorem, which says that
the equation A -- ot has a solution  in the smooth p-forms on M if and
only if the p-form ot is orthogonal (in a suitable inner product on E'(M))
to the space of harmonic p-forms (those for which Aq ffi 0). From the
Hodge decomposition theorem we will conclude that there exists a unique
harmonic form in each de Rham cohomology class. As another simple
application we will obtain the Poincar6 duality theorem for de Rham
cohomology and, from it, the Poincar6 duality theorem for real singular
cohomology. To prove the Hodge theorem, we shall give a complete self-
contained exposition of the local theory of elliptic operators, using Fourier
series as our basic too]. The eigenfunctions of the Laplace-Beltrami operator
and their use in a proof of the Peter-Weyl theorem are discussed in the
exercises at the end of this chapter.
THE LAPLACE-BELTRAMI OPERATOR
6.1 Definitions Recall (from 4.10(6) and Exercise 13 of Chapter 2)
that there is a linear operator * which assigns to each p-form on M an
(n --p) form and which satisfies
(l) ** ffi (--1)
We define an operator  from p-forms to (p -- 1) forms by setting
(2) a ffi (--1)""+1)+1"d *.
On 0-forms,  is simply the zero linear functional. The Laplace-Beltrami
operator A (Laplacian for short) is defined by
(3) A = d + d,
and is a linear operator on E(M) for each p with 0  p  n. We leave it

Laplace-Behrami Operawr 221
to the reader as an exercise to check that on Ee('), that is, on C ® functions
on Euclidean space I", the Laplacian is simply the operator (--l)
Also, it is a straightforward exercise to check that the Laplacian commutes
with ,, that is,
(4) *A = A*.
We define an inner product on the vector space E'(M) of-forms on M
by setting
(5) 0t,8) "=fff ^ "8 for t,/ 
and we denote the corresponding norm by lltll. It follows from (6) of Exer-
cise 13 in Chapter 2 that the bilinear form defined in (5) is actually symmetric
and positive definite. We extend the inner'products (5) for 0  p  n to an
inner prodtict on the direct sum  E(M) simply by requiring the various
E(M) to be orthogonal. -e
6.2 Proposition  is the adjoint old on _ E'(M); t/mr is,
(1) ( ,=,a) =
PROOF By linearity, and the orthogonality of the E(M), the proof
reduces to consideration of the case in which t is a (p -- 1) form and
8 is a p-form. In this case,
(2) d(= ^ *8) = d= ^ "8 + (-1),- ^
By integrating both sides over M and applying the special case of Stokes'
theorem contained in the corollary of 4.9 to the left-hand side, we obtain
(3)
Thus
(4)
Corollary
(5)
o = f(a= ^ "8 - = ^ *8) = (a,8) - (=,8).
A  e/f.adoim, har ,
(A=,8) = (=,A) (,   (M); 0    n).
Proposition   0  d oMy  d  0 d   O.
PR Cl¢afly & = 0 if d = 0 and  = 0. Now,
(1) <A,) = ((d + , ) = (,> + (d,d>.
us if A = 0, it follows at d = 0 and  = 0.

222 The Hodge Theorem
Corolla The only &rmonlc fnclon (f  O) on a compact, connected
oriented, Riemannlan manifold are the contant function.
TH HODGE THEOREM
6.4 Definition We shall let A* denote the adjoint of the Laplacian on
E(M). This operator is, of course, precisely A itself since the Laplacian is
self-adjoint on E(M), and usually we nake no distinction between A and
A*. However, this distinction will be important for the form of the following
definition.
We shall be interested in finding necessary and suicient conditions for
there to exist a solution o of the equation Ao  . Suppose that o is a
solution of Ao  . Then
(I) (A¢,?) ffi (,?) for all ? ¢ '(M),
from which it follows that
(2) (o,A*?) ffi (,?) forall ?¢"(M).
Now, (2) suggests that we can view a solution of A¢ =  u a certain type
of finear functional on (M), namely, o determines a bounded finem"
functional I on "(M) by
(3) tO) = (o,,0);
nd in view of (2), the functional I satisfies
(4) t(a*) = (,) for all  e '(M).
This view of a solution turns out to be extremely useful, for it will allow us
to bring various techniqu of functional analysis to hear on the problem
of solving A¢ = . We shall call such a finear functional a wet0c solution
ofA = . That is, a weak solution ofA¢ =  is a bounded linear functional
I: (M)-- [R such that
(5) /(a*) ffi (,) forall ¢P(M).
Later we will deal with weak solutions of a general partial differential
operator L dcfined on an open set in Euclidean space, and in place of A*
in (5), them will he the formal adjoint L* of L (see 6.24(3) and 6.31).
We have seen that each ordinary solution ¢ ¢'(M) of A¢ = 
dctermincs a weak solution by (3). It turns out that the major effort of this
chapter will be to prove a regularity theorem which says that the converse
of this is true; that is, each weak solution determines an ordinary solution.
The main step in proving this converse is to show that if I is a weak solution
of A == , thcn I is represcnted by a smooth form ¢ in the sense tlmt there
is a form ¢ ¢ E(M) such that (3) holds. That the form ¢ is then an
ordinary solution follows from the fact that

Hodge Theorem 223
(6) -
for all p  E'(M), which implies that
6.5 Regularity Theorem Let o ¢ E(M), and let I be a weak solution of
Ao , o. Then there exists ¢o  E(M) such that
for every/$  E'(M). Consequently, Ao --
We shall assume Theorem 6.5 for the moment as well as the following.
6.6 Theorem Let {,,} be a sequence of smooth p-forms on M such that
I1,,11 < c and IIAII < c for all n and for some constant c > O. Then a
subsequence of {,,} IS a Cauchy equence in E(M).
We will begin the machinery necessary for the proofs of Theorems 6.5
and 6.6 in the unit beginning with 6.15. We eventually return to the proofs
of Theorems 6.5 and 6.6 in 6.32 and 6.33 respectively. Meanwhile, we shall
assume the two theorems as proved and shall proceed to the Hodge theorem.
6.7 Definition We let
(1) {o, 40, 0}.
The elements of/P are called harmonic p-forms.
6.8 The Hodge Decomposition Theorem For each integer p with
0  p  n, It p is finite dimensional, and we have the following orthogonal
direct sum decompositions of the space E(M) of smooth p-forms on M:
(1) E,(M) H,
 d(E"l)  6(E +)  .
Coequently, the equation Aw   has a olution   E(M)  d only 
the p-form  i ortgonal to the ce of harmonic for.
pv If  we not finite mensional, then  wod contain an
infinite ohono quen. But by Theom 6.6, this ohonormal
sequen would ntn a Cauchy subuen, wch is imssible.
Thus  is fini mension.
It is sufficient to prove the domposition in the fit line of (1),
for the other two lines of (1) then follow from 6.1(3), 6.2, and 6.3.

Let wl, • • •, w be an orthonorma] basis of/. Then an arbitrary
form   E'(M) can uniquely be wr/Ren
(2)
where/Y lies in ()±, the subspace of E'(M) consisting of all elements
orthogonal to . Thus we have an orthogonal diret sum decomposi-
tion
(3) ;'(M) = (H') ± , H'.
The theorem will be proved by showing that (if')± =, A(E,). We
let/fdenote the projection operator of E(M) onto/P so that H(a) is the
harmonic part of
Now A(E') = (H)±. For if w  E ' and a  H , then
Conversely, we claim that (H) ± = A(E,). In order to prove this, we
first need the following inequality.
We claim that there is a constant c > 0 such that
(4) IlPll < c IIAPll for all   (H') .
Suppose the contrary. Then there exists a sequence 8 ¢ (/P)± with
linch ---- ! and IIAII - 0. By Theorem 6.6, a subsequence of the.
which for convenience we can assume to be {8} itself, is C.auchy. Thus
lim (8,W) exists for each o ¢ E'(M). We define a linear functional l
on E'(M) by setting
(5) l(w)  lim (/,W) for W ¢ E'(M).
Now I is clearly bounded, and
(6) l(Aq) =, lira (/,Aq) =, lira (Ap, q) =, 0,
so I is a weak solution of A/ =, 0. By Theorem 6.5, there exists
E'(M) such that l(o)=, (8,0). Consequently, 8-8. Since
11 ---- l and 8  (/P)±, it follows that II/11 ---- l and 8  (/-P')±. But
by Theorem 6.5, A8 =, 0, so 8  H , which is a contradiction. Thus
(4) is proved.
Now we shall use (4) to prove that (H) ±  A(Lr). Let a  (HP) ±.
We define a linear functional I on A(E ') by setting
(7) I(Aq) =, (a,q) for all q  E'(M).
Now I is well-defined; for if Aqh =, Aqs, then qh -- qs e H', so that
Ca, <Pt -- <P,) =' 0. Also ! is a bounded linear functional on A(E'), for let
<p e £'(M) and let o =, ? -- H(ep). Then using (4), we obtain

Hodge Tkeorem 22
By the Hahn-Banach theorem [27, p. 228], I extends to a bounded
linear functional on EP(M). Thus I is a weak solution of Aw
By Theorem 6.5, there exists (o • E(M) such that A¢o = o. Hence-
(9) (HP) ± --
and the Hodge decomposition theorem is proved.
6.9 Definition We define the Green's operator G: E(M) -* (H) ± by
setting G() equal to the unique solution of Aw   -- H() in (H) ±.
We leave it to the reader as an exercise to prove that G is a bounded self-
adjoint linear operator which takes bounded sequences into sequences
with Cauchy subsequences.
6.10 Proposition G commutes with d, 6, and A. In fact, G commutes
with any linear operator which commutes with the Laplacian A.
poor Suppose that TA  AT with, say, T: E(M)-* E*(M). Let
rH,± denote the projection mapping of '(M) onto (/P)±. By
definition, G  (A I (H)±) -1 ° rH,±. Now, the fact that TA  AT
implies that T(HD c /p; and since (HD ± -- A(E'), it implies also that
• T((/P') ±) c (Hq) ±. It follows that
(1) T o r,)± ---- rH)± o T,
and on
(2) To (A I (He) ±) ---- (A I (H')±) o T,
and hence on (H) ±,
(3) To (A I (H,)±)-' ffi (A i (m)l)-,o r.
It follows from (1) and (3) that G commutes with T.
6.11 Theorem Each de Rham cohomology class on a compact oriented
Riemannian manifold M contains a unique harmonic representative.
ptoor Let  be an arbitraryp-form on M. From the Hodge decomposi-
tion theorem and from the definition of the Green's operator G, we have
(1) o  d6Go + 6dGo + Ho.
Since G, by 6.10, commutes with d, we have
(2)
Thus if  is a closed p-form,
(3)  ffi d6G + H,

226 The Hodge Theorem
so Ha is a harmonicp-form in the same de Rham cohomology class as is
a. If two harmonic forms al and  differ by an exact form d/, then
we have
0 =' dp+ (1 --
But dp and (1 -- ) are orthogonal since
(a, al - =) = (, sa, - s= - ,0) = 0.
Thus d = 0 and = = . Thus there is a unique harmonic form in
each de Rham cohomology class.
Corollary The d Rham cohamology groups for a compact, orlentable,
dierentiable manifold m,e all finite dimensianal.
,aooF Any differentiable manifold can be equipped with a Riemannian
metric (Exercise 23 of Chapter 1), and so the corollary follows immedi-
ately from Theorem 6.11 and from the finite dimensionality (6.8) of the
spaces H" of harmonic forms.
6.12 Let M be a compact, oriented, differentiable manifold of dimension n.
We define a bil/near function
(1) H.,tM) x H."T',(O'-" R
by sending
(2) ({},{}),-' j ^ ,
where 9 and 0 are clo fos psenting e homolo d {9} in
He a(M) and {} in H(M). Oboe at the bifine map (2) is weD-
defined. For exple, if 9t is other pnfive of e de Rh cls
{9}, then 9t  9 + d for some fo , d by e si se of Stoke'
theom wch is nined in e rolla of 4.9,
Obsee so from i difion at the bilinr function (2) den on e
oentnfion on M.
6.13 Theorem (Poincar6 duality for the de Rham ¢ohomology of
a compact oriented n-dimensional numifold B4) The bilinear function
6.12(2) is a non-singular p.airing and consequently determines lsomorphisrns
of l-l(M) with the dual space of les(M):
(I) - H(M)  (H.,(M))*.

Some Ca/culus 227
!oo1: Given a non-zero cohomology class {?)cHez(M), we must
find a non-zero cohomology class {0} ¢ Se(M ) such that ({?},{0}) ¢
0. Choose a Riemannian structure on M. We can assume, according
to 6.11, that ? is the harmonic representative of {?}. Since the cohom-
ology class {?} is not zero, ? is not identically zero. Since .A - A.,
it follows that .? is also harmonic, and therefore closed by 6.3, and
so .? represents a cohomology class {.?} ¢ H'(M). Now
0.
Thus the pairing 6.12(2) is non-singular, and the isomorphism (1)
follows from 2.7.
Corollary If M i a compact, connected, orientable, differentiable manifold
of dimension n, then  (M)  .
6.14 Remark The real continuous singular homology groups H,(M;II)
are defined just as in 4.16, with the exception that all continuous simplices
in M are allowed, not just differentiable simplices. The isomorphism 5.37(1)
holds for the continuous case exactly as for the differentiable theory. By
combining 6.13 with the de Rham theorem 5.36, with the canonical isomor-
phism 5.32(5) of the differentiable with the continuous real singular cohom-
ology of M, and with the isomorphism 5.37(1) for the continuous real
cohomology and homology, we obtain the Poincar duality between the real
singular cohomology and the real singular homology of M:
(1) (M;)  ,,_,(M;).
SOlVIE CALCULUS
We now begin to develop the machinery necessary for the proofs of Theorems
6.5 and 6.6.
6.15 Notation We shall be using multi-index notation 
where the  are integers. We let [l denote the ordinary Euclidean norm of
; that is,
and if the  are all non-negative, then we let
(2) [] = t + "'" + ..
If  and  are both n-tuples of integers, then
(3)   t • • • -, where we set 0 °  1.

228 The ttodge 77eorem
We shall use x, for the ith anonical coordinate function on I-, and let
(4) x - (xl,..., xD.
The xth derivative operator D" is defined for each n-tuple x of non-
negative integers by
where i  x/. (The addition of the factor of i will be convenient later.)
We let ' denote the complex vector space consisting of C ® functions
defined on [" which have values in complex m Sl,-ce C" and are periodic
of period 2" in each variable. I would suggest that the reader assume m to
be 1 for the first reading of this section since that case contains all of the
essential ideas and the computations are somewhat simpler. We will arrange
the notation so that it is essentially the same for general m. One note .of
caution concerning the notation is this. If y, p c C', then y. p means the
Hermitian product ydx -l- "" -i- y,/,. If m happens to be 1, then
means y/. The associated norm is denoted by
If ?, V c, then ?., is the complex-valued function which is the
Hermitian product of ? and ,; that is,
(6) .  = ,- +... + ,-.
We let Iwl denote the real-valued function
(7) Iwl  (w"
If ?   and if f is a periodic complex-valued C ® function on
periods always 2r), then ?f shall denote the element of  whose ith com-
ponent function is ,f; that is,
Let Q  I- be the open cube
(9)  = (t,  : o < x,(t,) < 2, t =  .... ,
We shall be introducing a number of different norms on '. By Ilwll we
shall mean the ordinary/ norm of , over ,
(10) ll,t, ll = (2)  ,t," w) ,
and (V,?) shall denote the/. inner product,
(it) (,t,,q) =  w"
The norm llwll® shall denote the uniform norm of
(12) llwll® = sp l,t,l.

,ome Calculu 229
6.16 Some Facts about Fourier ,Series If ? and if 
(1,..., ) where the , are integers, then the th Fourier coefficient
?e  C' is defined by
where x. e  xtl + " "" + x.
First we are going to show that the Fourier series  ?re '*'t of ? converges
uniformly to ?. Let an integer k > 0 be given. By integrating (1) repeatedly
by parts, differentiating ? and integrating • -*t, and observing that the
boundary terms drop out since the integrand is periodic, one sees that there
is a constant c depending on ? and its derivatives up to order at most
2nk such that
(2) I?el  (]'I )"--'--- for all  • 0,
where ]Ie denotes the product of all the non-zero e. It follows that there
is a constant ct such that
(3) I?el  c, for all .
(l + I1') 
Consider now the question of convergence of the series
If we let
(4) S _-- { _-- (et .... , e): t<_<max [[ ---- j},
then the number of elements of S is at most 2n(2j + 1) "-t, and for each
S, I1  j, so that
es 2n(2j + 1) "-" ci.__,.
(5) s,  ,(1 + I#1=)  (1 +
forj  1, whe c is a constant dending only on n. Consequently,
® 1
(6)
 (1 + I,1) ' ,-x -
and so the series  (1 + I1) - converges for I + 2k -- n > 1, 6r in other
words, for
(7)
where [/2] denotes the greatest integer less than or equal to ]2. It would
be an interesting exercise for the reader to reach the same conclusion by
using an integral test.

230 The Hodge Threm
It follows from (3) that if we take k >. In/2] + I as in (7), then the Fourier
converges uniformly to some continuous function . We olalm that q  ?.
This is of course due to the completeness_ of the trigonometrio system and
may be deduced as follows from the Stone-Weierstrass theorem [13], [27].
Let p m ? -- q, and let t e be a trigonometrio polynomial; that is,
t is a finite linear combination of terms of the form ale*'t for a t e C '.
Then ince ? and q have the ame Fourier coefficients,
O.
Now if • > 0 is given, then by thc Stonc-Vciersass theorem thcre is a
trigonometric polynomial t ¢@ such that llw -tll® < e. Thus, using (9),
we see that
Consequentty IlVll < e. But • > 0 was arbitrary and F is continuous, so
F = 0. Thus the Fourier series of a periodic C ® function ? converges
uniformly to ?,
(11) (x) = . ¢pee '-'e.
It follows from integration by parts that the th Fourier coefficient of
D is x""" ,"e. Thus
(12)
From (11) and the orthogonality of the trigonometric system, we obtain
the Parseval identity:
Applying (13) to D*?, we obtain
(14) II D'qll I   .-iqel I,
It follows from (14) that given a non-negative integer t, there is a constant
c greater than zero and depending only on t and n such that
(15) c(1 + I  l  )'lq,el ' < t.  ollD'  ell ' < + I  l')'lq,el

Some C.a/csdus 231
6.17 The Sobolev spacea H, Let $ denote the complex vector space
consisting of all sequences of complex vectors in C ' indexed by n-tuples of
integers   (x, .... ,,). Thus if u e $, u  {u t} where  runs over all
n-tuples of integers and where each u t e C '. For each integer s (positive,
negative, or zero) the Sobolev space H, is the subspace of $ defined by
(1)
It follows from the Schwarz inequality that
hence the left-hand side of (2) is finite whenever each member of the right-
hand side is finite. We can therefore define an inner product on  by.
The associated norm is
It also follows from (2) that (u,). exists if u e H, and  e H., where
(t + t')/2
Since H. is simply an II-space in which the measure space is the set of
all n-tuples of integers , and the measure is the counting measure weighted
by (1 + [ls) ", it follows that H. is a Hilbert space.
We define a linear transformation K * on 8 for each integer t by setting
(5)
Finally, we identify ' with a subspace of $ by associating with each
? ' its sequence of Fourier coefficients {?¢}. In view of 6.16(12) we can
extend the derivative operator D" from ' to all of $ by setting
The inequality 6.16(3) together with 6.16{7) shows that the more dilferen-
tiable a function is, the greater the order s of the Sobolev space H. in which
its sequence of Fourier coefficients lies. In particular,
Moreover, P is dense in each H, since each u  H, for which all but a finite
number of the u¢ are zero belongs to P. We consider elements of the Sobo]ev
spaces H, as formal Fourier series or "generalized functions." A fundamental
]emma due to Sobo]ev says that if u  H', for sufficiently large s, then the
formal Fourier series determined by u actually converges to a function with
a certain number of derivatives depending on s. This lemma will be one of
the key steps in the proof of the regularity theorem, for it will allow us to
conclude that a generalized solution of a partial differential equation which
belongs to a sufficiently high Sobolev space is an actual solution.

232 The Hodge Trem
Some salient features of the H0 spaces are collected in the following
theorem, which is divided into parts (a)-(j).
6.18
(b)
(c)
(d)
(0
Let s be a non.negatire integer. Then there are constants c and c',
depending at most on s and n, such that
(l) c tlil0  : IID'qfll  c' I111, for all 
{]-O
Mor¢orer, for the cae •  O, we tly e the elity
(2) 11 = IIllo for !  .
if t < , then Ilull,  Ilull,, o H,  H,.  the ion of the
• pace orer all inter • i a pace of 8, which we denote by
  a dense pace of H, for each .
K'   iometry of H, onto Hu with inverse K -,
(3) llull,  IIg'ull,, ;
d K  maps  to . If    d t  O, then
(4)
Moreover, for all s and t
(5) (u,v)•  (u,K'v),_,  (K'u,v),_, for u, v  H,.
Schwartz Inequality If u  H,+, and v  H,_,, then
I(u,v>01 < Ilull,+, IIvll0-,.
If u  H,+, , then
llull,+, = sup
,,.-,
(g) Peter-Paul Inequality Given integers t" < t < t" and • > O, there is
a constant c(e) > 0 such that
Ilull,' < • Ilull,-' + c(e) Ilull,,'
for all u  H:. (We refer to this inequality as the Peter-Paul inequality
in view of the comparison with part (b), although the morality has
been twisted around. Rather than robbing Peter to pay Paul, here we
are paying Paul (Hull,.) in order to rob Peter (llull,-).)
(h) D" is a bounded operator from H,+[.] to H•for each s; indeed,
IID'ull• < llull,+[,] for all u e H,+[. 1.

Some Calculus 233
(i)
(j)
Let w be a C ® complex-valued periodic function on n. Then given
an integer $, these are positive integers c and c', with c depending only
on s and n, and c' depending on $, n, and on  and its derivatives,
such that
(6) IIoPll, < c IIll® IIPlJ, + c' IIPll,-, for ?
In particular, there 15 a constant c" depending on , $, and n such that
(7) IIoPII, < c" IIPll,,
$o multiplication by  extends by continuity to a bounded operator on H
Let  be a C  complex-valued periodic function on [. Then given an
integer s, there is a positive constant c such that
(8) I<u,v), - <u,v>,l < c(llull, Ilvll,- + Ilull,_ Ilvll,)
for each u, v  He. For the case s , O we have
(9) (ou,v)o  (u,V)o.
proof The inequality (1) follows from 6.16(15) and the fact that
whenever al ..... a. are positive numbers, then
The equality (2) is simply the Parseval identity 6.16(13).
Part (b) is obvious. The fact that 6.16(3) holds for each integer
k > 0 implies that {p¢}  H e whenever p ca ), and as we have already
indicated in the remarks immediately preceding the theorem,  is
dense in H, since each u  H, for which all but a finite number of the
are zero belongs to a ).
The identities (3) and (5) in (d) are obvious; that the inner products
are all well-defined follows from the remark immediately following
6.1"/(4). Let cp ,q). We have Ktcp ---- {(1 + Ilt)qp¢}. It follows from
6.16(3) and (7) and from the fact that there exists the inequality
that the series
and all its formal derivatives
(13)  D'(1 + I[t)tpce e''¢ ---- '(1 + I[t)'te
converge uniformly. Thus (12) converges to a periodic C  function and
therefore is an element of #. Hence K  maps  into #. For (4),
simply observe that the th Fourier coefficient of the right-hand side
equals (1 +

The Hode Theorem
The Schwartz inequality (e) is none other than 6.17(2). It follows from (e)
that
To prove the equality in (f), let v  KZu. Then by (d),
To prove the Peter-Paul inequality (g), first observe that for any
positive number),,
(14)
since either), or 1/), is greater than or equal to 1. If in (14) we let
we obtain
from which the Peter-Paul inequafity follows with
The inequality in {h)follows from the inequality {II).
In part (i), the inequality (7) follows immediately from part (b) and
inequality (6). To prove (6), we first consider the se in which
s : 0. Let q  . Then by applying {I), we obtain
(.)-o [a)-o
At the third stage we have used the fact that D'o? -- oD'q onlycontains
derivatives of q up to order [a] -- 1. For $ < 0 we have
= {K-•oK•,K•ap)o + ({oK -• --
I [a)--O " 0 1

Some Calculus 235
where the a s are combinations of derivatives of ¢o. Now. by the case
$ : 0 of (i) we obtain
< (c Jloll® II/Oqll_. + k'
: (c floll® H+II, ÷ k' fl +g,_) IIo+ll,.
As for the last term in (15),
I ,-s*--t % I --t
 oonst  flD'K', iIK'oPII_.
 mnst Y IIK°qii,+(.3 IIK°oPIl_o
< ot HK°qll_,_t liK°ePll_0
< eot iiqil,-t liePllo.
Thus for s < 0, (6) follows from (15), (16), and (17).
-- It is suicient tO prove (8) and (9) in part (j) on the dense suSspace
of Ho. Observe that (9) follows immediately from the fact (1) that the
/.-norm is identical with the Sobolev norm il lie on.e. Let q,
We consider the case of (8) in whic5 $ is negative. Using part (d)and
equation (9), we obtain
It follows as in (17) that
I<¢oq.W)o -- <q.cW)oJ  const IIqll, IIwII,-t-
By symmetry we also have
I(c,Wo -- (q.cP)ol  const ilv, llo il
Thus (8) is proved for $ negative. The proof is similar for $ positive.
The proof of Theorem 6.18 is complete.
6.19 Difference Quotients If q  , then the th Fourier coeicient
of the translate q(x + h) of q by an element h  ' is e'tqt. Thus if
u  8 and h  ', we define the translate ofu by h to be the element

236 The Hodge Theorem
The difference quotient of u determined by a non-zero h is the element
Ihl I 
So if  e , then T()(x)  (x + h) and
(x)  (x + h) -- x)
Oe that if u e H,, then
(3) II T(u)L ffi Ilull,,
so T is an isomet on H,. Thus, in paicular, if u  H,, then also u   H,
for each h. It follows from the inequality
 4 sin*(h • )    (1 + I1 )
ihl   ihl 
tt ff u e H, then the u *  uformly unded in the s-norm. In fact,
(5) I111,  Ilull.
We sh! nd the conve of s, as follows.
6.20 I.mn Let u e He, and assume that there is a constant k such that
Ilull, < k for all non-zero h  ". Then u  H+I.
PlOOV For each positive integer N we let u s be the element of He
obtained by truncating u at N; that is,
(uN)==lu if II<N
(1)
[0
otherwise.
We need only prove that the IluL+l are uniformly bounded. Let
(e,..., en) be the standard orthonormal basis of n, and let h , te.
Then
I e'''- IV-I:"-ll'--.l,l" as t0.
(2) lhl t
Since there are only finitely many  with I I < N, and since by hypothesis
lhl
it follows from (2) that

Some Calculus 237
Thus Ilu.ll,+l "= - (1 4- Ilt)'+lluelt  nk t 4- Hull, t, so the Ilull,+a are
• Ii.v
umformly bounded, and consequently u  H,+z.
6.21 Associate with each u  H_® the series Y_ uteZ't. It will be of funda-
mental importance in what follows to know when this series converges, and
(if it converges) how differentiable its limit is. The answer is supplied by the
following lemma due to Sobolev.
6.22 Sobolev I.emma If t  [n/2] + 1 and u  H, then the series
Y_ ue tz' converges uniformly. Thus each u  Htfor t  [n/2] + 1 corresponds
to a continuous function.
PRoo It is suicient to demonstrate that the series converges absolutely,
Y_. lull < o. Now,
E lull = ,,(1+ I1")-'/'(1 + I1") '/" lull
I1.."
Thus the result follows from 6.16(7).
Coronary (a) If u  lit where t  In/2] + 1 + m, then D'u  Y_ 'ute'*'
converges uniformly for [0]  m. Thus each u  Htfor this range oft corre-
sponds to a function Y_ utei*'t of cluss C".
PROOF With uHt for t:[n]214-14-m and with [0]m, it
follows from 6.18(h) that Du  Ht_t], where t -- [0] : [n/2] + 1.
Thus it follows from the Sobolev lemma that Y_ 'ute*t converges
uniformly. Now this series is the 0th "formal" derivative of the series
 ute '. Thus Y_ ute*t is of class C".
Corollary (b) From the proof of 6.22 it follows that if t  [n/2] + 1,
then there is a constant c  0 such that if   , then
() IIqll®  c IIqllt.
,,lpplying (!) to D"q and using 6.18(h), we obtain
(2) IID'ell® < c Ilellt+,,].
6.23 Rellich Lemma Let {u } be a sequence of elements of Ht with
Ilull  lo f s < t, then there is a subsequence of {u } ),'hich converges in
PRoor By assumption,
(1)  (1 + Il')' lull" < 1.

235 The Hod&e Theorem
For each fixed $, elements Of the sequence (l(1 + I#l=)*=ull) are all
bounded by 1, so the sequence ((1 + I#l=)'l=u} has a convergent sub-
sequence in C '. By the usual diagonal process, one can select a sub-
sequence (u ,) such that the sequence (1 + Il=)'Itu' ' converges in C '
for each fixed . We claim that (u ,) is Cauchy, and therefore convergent,
in H, if s < t. Let • . 0 be given. Now,
(2) Ilu"- u"llo = - . ( ÷ I1=)'-'(1 ÷ I1')' lug' - u'l =
The second sum in (2) is bounded by
('--')  (1 + I'l=)'(lu'l = +  lul'l lu.'l +
IliON
wch is 4 (') in view of (I). Sin • -- t  0, 4 (') n 
me !  /2 by king N ! enou,  N = No. e fit
s in (2) is then und by
and sin e  oy a fi humor of  in ts s d n
the uen (I + II=)' ' nv for h  , e is a
const J > 0 such that j, and jt a r  J, tn (3) is I
 /2.  forj,,jt > J, we have Jl' -- .,s < , d the prf
is mpl.
6.24 DefinlUona A (linear) differential operator L of order ! on the
C'-valued C ® functions on IR,, consists of an m x m matrix (L,) in which
and the a are C ® complex-valued functions on IR,,, with at least one
a  0 for some l, j and for some • for which [] =. 1. A differential operator
L is a periodic differential operator, or an operator on , if, in addition,
the a are periodic functions.
Let L be a periodic differential operator, and let q   with component
functions qt ..... q,,. Then
(2)
It follows from integration by parts that if we define the operator L* on ' by

Some Calculus 239
so that the ith component of L*q is given by
(4) (L%),  Y Y a,),
then in e  inner pru on  we have
(5) (,>
for ,   . L* is ! the fo joint of L. e word "form" is
ud he to emphi at L* is not e joint of L on a Hiln se.
It is mply
6.25 Propmitton Let L be a partial differential operator on  of order l,
and let s be an nteger. Then there are positive constants c, k, and c', where c
depends only on n, m, l, and s, where k is a bound on the absolute values of
the coe, Ocients of the highest order terms n L, and where c' depends on n, m, 1,
s and on all the coe, Ocients of L and their derivatives up to order 1, such that
(1) JJLq8o  ck IIqll-s + c' JJqll-s-z
for all q  . In particular, there  a constant c" such that
(2) IILqfllo  c" IIqll,+s
for all e  , so L extends by continuity to a bounded operator from H,++
to Hofor each s.
pRoov The inequality (2) is an immediate consequence of (1)-and
6.18(b). As for (1), the case m = 1 (in which elements of  are C x
valued functions and the operator L is a single partial differential
operator a*D" rather than a matrix) follows immediately from
6.18(h) anl (i). The inequality (1) for general m follows from the cse
m = 1 and from the inequality IILqllo  const Y_ IILaqll. where
q = (% ..... q,) s , and the constant depends only on m.
6.26 Remark If L is an operator on  of order 1, and if o is a C ®
complex-valued function on ", then the operator M =, oL -- Lo, where
Mq = o(Lq) -- L(oq), is of order at most l -- 1. Consequently, given s,
there is a positive constant such that
6.27 Lemma If w is a real-valued periodic C ® function, and L is a differen-
tial operator of order I on , then there ts a positive constant such that
(1) I(L(+o'u), Lu). -- IIL(u)ll.l  +omt (llull.+, Ilull.++-.)
for all u  H+t.

100 i(L(u), L>, -- (L(ou),
and (1) rollows b applng 6.1g), 6.25(2), and 6.1g( to
 d applang  hw anui¢, 6.26(1), 6.25(2), and 6.1g(
to the last two tes.
OPERATORS
6.28 Definition Let L be a partial differential operator of order L
We write L as
(1) L , P(D) +... 4-
where P(D) is an m x m matrix each entry of which is a differential operator
 a.D', homogeneous of order J, and where the a, are C  complex-
valued functions on *. We let P,( denote the matrix obtained by sub-
stituting " for D" in P(D), where  , (1 ..... ,) is a point in n*.
L is said to he elliptic at thepoint x  * if the matrix P() is non-singular
at x for each non-zero . L is elliptic if it is elliptic at each. x. Observe that
ell/pficity is a condition on the highest-order part of L only. Observe also
that L is elliptic at x if and only if
(2) L(u)(x)  0
for each C"-valued C ® function u such that u(x) 0 and each smooth
real-valued function q such that q(x) ffi 0 but de?(x)  O, since for each such
 and u,
(3) L(uXx) - P( l)Xu)(x)  P(dl.)(u(x)).
The advantage of this criterion (2) for ellipticity is that it generalizes to a
coordinate-free definition of ellipticity on manifolds, as we shall see later.
.The basic analytic property of an elliptic operator that we shall need is
the following.
6.29 Fundamental Inequality Let L be an elliptic operator on  of
order l, and let s be an integer. Then there is a constant c > 0 such that
(1) Ilull.+, : c(llLull. ÷ Ilull.)
for all u e H+,.
nOOF It is sufficient to prove (!) for all p  9 . The proof consists
of everal parts. We first consider the caze of an elliptic operator/,
on  with constant coefficients, which consists of the leading term

Elliptic Operators 241
P(D) only. If u ¢ JR" with u O 0, and if"  O 0, then since P() is
non-singular, we have [Pz()ul* > 0. It follows from the compactness
of the unit sphere in JR,, that there is a constant c > 0 such that
IP()ul t > c
for all u and $ such that lul  I1  1. From this it follows that
(2) Ie,()ul' > el 1 'z lul'
for all u and  in I n. Thus for q  , it follows from (2) and the fact
that Lo has constant coefficients that
(3)
Hence
(4)
(ll,:plh, + IIqll.)' > IIqll.' + IIqfl,'
> : lql'(l + lIt)'(l + oonst I1 '))
> co,,t  Iqel * (1 + I1') '+
Secondly, consider a general periodic elliptic operator L of order 1,
and let p  IR,,. We shall prove that there is a neighborhood U of p
such that (1) holds for all q   with support in U. (By a slight abuse
of terminology we say that the support of a periodic function q lies
in U if the support of q lies in the union of U with all of the periodic
translates of U.) Let L0 denote the constant coefficient elliptic operator,
homogeneous of order l, determined by the highest-order part of L
at the point p. Then it follows from (4) that for each q  ,
(5) II qll.+, < eonst (llZqll° + II qllo)
 const (llLqllo + II(L0 - L)qllo + IIqll°).
Let k denote the constant in (5). Then choose a positive e smaller
than l/(2ck), where e denotes the constant e of 6.25. On a small enough
neighborhood of p the coefficients of the highest-order part of L0 -- L
are less than e in absolute value. Let/' be a periodic operator agreeing
with L0- L on a possibly smaller neighborhood U of p and with
coefficients in the highest-order part everywhere less than e in absolute
value. Then it follows from (5), 6.25(I), and the choice of e, that for
an element ?   whose support lies in U,
IIqll,+, < const (llLqllo + II/qllo + IIqllo)
 const IILqll, + ½ llqll.+ + const II qll,+,-i + const IIqllo.

242 e Hodge Theorem
Applying the Peter-Paul inequality to the term lJ qflJ,+-s, we obtain
fl qfli.+: : ,n=t iiLqflio -I- ! ii qfli.+, -I- ot
which proves (I) for these q.
Let T" be the torus obtained as the quotient of " by the lattice
consisting of all points 2', where  is an n-tuplc of integers. The open
sets U obtained above for each p e " project to an open cover of
Let Us ..... Ut be a finite subcovcr, and let wl, .... wt be a partitio.n
of unity fitting this cover, of the special form
(6) Y_ ,? : I.
This is easily arranged--see the proofs of 1.11 and 1.10. Now consider
the ¢o, as periodic C ® (real-valued) functions on iR". Let ?
Then by (6), and by 6.180) and Q'),
Now sin 9 h sup in one of the sm on ms U obmin
avc, and sin thc  oy finitely many of c , thc a
nstants such that the last display line avc is
 mt  IIfl + t I111. = + t I111 flll.+t
 t  <,'),>. + mt flll + mt flll: I111
' (by 6.27)
: mt IIfl + mt flfl. * + mt BII.+ IIll-x
 t iill + t I111.* + } iill+ + t liii,-t
 t ilil + t liii.' + t liii + t IIil.
(Sy the Peter.Paul inuity).
It follows that (1) holds for all ep e  and hence for all u e H.+g.
6.30 Theorem (Regularity for Periodic Elliptic Operators) Let L
be a periodic elliptic operator of order L Assume that u e H_® , v e H , and
(I) Lu : v.
Then u  H+ .

.Reduction to the Periodic Case 243
PROOF It is sufficient to prove that if u ¢ H, and v ,/..u ¢ H-+I,
then u  H,+I. Let h ¢ In with h  0, and let L a represent the operator
obtained from L by replacing each coefficient • by its difference quotient
(x + h) - (x)
lhl
Then it follows f,r ? ¢ , and hence by continuity for all u ¢ H_®,
that
(2) L(u )  (Lu)  -- L(T,u) (cf. 6.19).
It follows from (2) and the Fundamental Inequality 6.29(I), that
(3) Ilull, : cont IIL(u)ll,_, ÷ cont Ilull,_,
: const II(Lu)ll,_ + const IIL(Tu)II,_ ÷ const
Now since the coefficients of the operator L are periodic C ® functions,
their difference quotients are uniformly bounded, so that
(4) IIL(Tu)II,_  const IIT,(u)ll,
where the constant does not depend on h. It follows from (3), (4),
and 6.19(3) and (5) that
Ilull0 : cont IILull,-,÷x ÷ const
where the right-hand side is independent of h. Thus, by 6.20, u  H+,
and the theorem is proved.
REDUCTION TO THE PERIODIC CASE
6.31 Remarks Before beginning the proof of Theorem 6.5 we need to
establish some convenient notation and to make a few observations.
We are going to let C ® denote the set of all complex m-space valued C ®
functions on It,,. C' will denote those of compact support, and C(V)
those whose compact support lies in V. By the L2 inner product on C
we shall mean
(I) <,> -- ¢---) . u. ,
where u. v as before denotes the Hermitian product u + • - • + u=t'7.
Let V be an open set in I* with V contained in some 2r cube. Then by
extending periodically we can (and do) identify C(V) with a subspace
of . Observe that the L2 ihner product on C(V) agrees with the L
inner product on C(V) considered as a subset of , which in turn agrees
with the Sobolev inner product II lle.

244 The Hodge Theorem
Now suppose that L is an elliptic partial differential operator of order !
on C ® (no assumption of periodic coefficients). L has a formal adjoint
L* on C, with respect to the ordinary L, inner product on C, obtained by
integration by parts. Just as in 6.24, if L .= (L) where Lo
then L* .= (L) where L .=  D',. L* is also a differential operator
of order l.
Now let p  I-. Then there is a sufficiently small neighborhood V of
p and a periodic elliptic operator Z such that Z agrees with L on V. For let
Le denote the constant coefficient operator determined by L at p. Then
since L is elliptic at p, there is some • > 0 such that any operator whose
coefficients are everywhere within • in absolute value of the corresponding
coefficients of Le is elliptic. So let U be a neighborhood of p, small enough
to be contained in some 2r cube Q, on which the coefficients of L differ from
the corresponding coefficients of/. by at most e; and let V = V = U.
Choose a C ® function ? with 0 : ? : 1 such that ? is 1 on V and has
support in U. Then the operator
is elliptic on all of I- and clearly can be extended from Q to be a periodic
elliptic operator Z which agrees with L on V.
We shall need the following slight extension of the formal adjoint property
of L*. Let u  H,, and let ?  C'(V). Then
(2) <Zu, qo ,= <u,r*qo.
For let p --- u in II II, with p  . Then
<Z,q>0- <Z,q>- <L,q>
= <p ,L* ?> = <p ,L*
so that
I<.u.q>o - <u.L*q>01 = I<Z:(u - V,). q>0 - <u - V,. L*q>01
 II/:( -- V,)II,- IIqll-.+ + I1 -- V, II. IIL*qll-.
 const I1 -- VII. IIqll-.+ + I1 -- VII. IIL*qll-..
which converges to zero as j--- .
Let V, as above, be an open set whose closure lies in some 2r cube.
Let u and  belong to H,. Then we shall say that u and  are equal on V if
(3) <u - u. q>0 " 0
for all ?  C (V). We shall say that a perfodfc opfrrfor  hi$ support in V
if the coefficients of L belong to C(V) c ,. Now, if L has support in
V, and if the elements u and  of H, are equal on V, then
(4) Lu , L.
For by 6.18(r), it is sufficient to prove that

Reduction to the Periodic Case 245
(5) <L(u - v), ?>0 = 0
for all ? ¢ . But applying (2) (with/' -- L), we obtain
<L(u -- v), ?>o - <(u - v) L*?>o
which is zero by (3) since L*? ¢ c(v) c @.
6.32 Proof of the Regularity Theorem 6.5 We are going to restate
Theorem 6.5 in slightly different notation, adapted to the local problem in
I- to which the theorem will immediatdly be reduced. We shall use
( , )" to denote the inner product 6.1(5) on E'(M). We shall prove:
(1) Given a C® p-form f on M and a bounded linear functionai i': E(M)
such that/'(A*?)  (f,?)" for every ?.¢ E(M), then there exists a C ®
p-form u on M such that i'(t)  (u,t)" for every t ¢ E(M).
We note that the last statement of Theorem 6.5, which says that Au
is an immediate consequence of (l), since, as we have already observed in
6.4(6), (Au,?)'  (u,A*?)' =:/'(A*?)  (f,?)' for all ?  E(M).
We now reduce Theorem 6.5 to a local problem. Let U be a coordinate
patch on M with coordinate map 7 such that 7(U) I n. Via this co-
ordinate system, differentiable p-forms become vector-valued functions from
I'to I'cC" wherem. (;). Weadoptthenotation of 6.31. Sovia
the coordinate system (U,,), p-forms on M yield elements of C ®; and in the
reverse direction, each element of C extends by zero to a complex-valued
p-form on all of M. The Laplacian A induces a partial differential operator
L of order 2 on C ®. The basic fact that we shall need concerning L is that
L is an elliptic operator. This we assume for the moment and shall establish
in 6.35. We let L* denote the formal adjoint of L with respect to the/.
inner product on C.
We extend the inner product ( , )' to complex-valued p-forms in the obvi-
ous way, so that if ul, uz, vl, and vz are real-valued p-forms, then
(u + iu, v + ivy)' ffi (u,v)" + (u,v)' + i((uz,vl)" - (u,v)').
Then by transferring this to Euclidean space we obtain another inner product
( , )' on C induced from the inner product on complex-valued p-forms on
M. It follows from the observation that both the/ inner product
and the inner product ( , )' on C are integrals of pointwise inner prod-
ucts, that there exists a matrix A of smooth functions on I,,, Hermitian
and positive definite at each point, such that
for all ?, o  C.

246 The Hodge Theorem
The adjoint of L on C' with respect to ( , )' is simply A* restricted to
C. We claim that for ?
(3) L*?  AA*A-I?.
Indeed, for arbitrary p e
<L* ?,o>
 <A*l-lq, ,>'  <lA*l-lq, ,>.
We extend the linear functional 1': E'(M)---IR complex linearily to
complex-valued differential forms, and we define a complex-valued linear
functional i on C' by setting
(4) i(?) = i'(/1-?).
We claim that i is locally represented by a C ® function. Precisely, we shah
prove:
(5) If p e I n, then there is
such that l(t) = (u,,t)for every t  C(W,).
Pirst we show that (5) implies (1). It follows from (5) that for each
p, q  IR n, up I Wp  W,  u,I Wp  W, since both up and u, have the same
Lz inner product with all elements of C(Wp  W,). Thus the up piece
together to give u e C ® such that u I Wp  upl W, for each p e IR. Now
let {?,} be a partition of unity on IR  subordinate to the {Wp}. Then if
t  C', l(t)  , l(?d)  , <u,?d)  <u,t). Now if ? is a smooth p-form
on M with support in U, then I'(?)  I(A?)  (u,A?)  (u,?)'. By the
same argument as above, the u's for various coordinate systems on M piece
together to form a C ® p-form u (necessarily real-valued) on M such that
l'(e?)  (u,e?)" for every q e E'(M). Thus we have reduced the proofof(1)
to the proof of (5), which follows.
Let p  IR  be fixed, and let Q" be some open 2,r cube containing p.
Choose an open set V such that p ¢ V  V  Q', and let
(6) T=
First of all, we observe that " is a bounded linear functional on C(V).
For since V is compact, the matrix norms II/.-111 have a maximum as x
ranges over V, and thus, using the fact that i" is bounded, we obtain
I)'(q)l  Ii(q)l  Ii'(/l-q)l < corot II,,l-q H'
= corot (</l-q,l-q0') uz = corot <q,/l-q0
< omt (II qll ll/l-lqfll) = < omt llqfll
for all ?  C°(V). Second, we observe that it follows from (6), (4), (3),
(2), and (I) that
(7) T(L*) = I'(/I-L*?) = i'(A*/I-?) = (f,/l-?>" =

Reduction to the Periodic Case 247
for all q ¢ C0 (V). Thus 'is a weak solution of Lu.ffif
Since " is bounded, " extends to a bounded linear functional on He.
It follows that there is an clement t  He such that
(8) '(t) = (t,t>0 for all t  H0.
Our task is to show that on a small enough neighborhood of p, the element t
agrees with an element of P.
Choose a neighborhood O0 ofp with 00 = V small enough so that there
exists a periodic elliptic operator/' which agrees with L on O0 (cf. 6.31).
Choose a neighborhood O ofp such that O = O0, and then choose a sequence
of neighborhoods O. of p such that  = O. and  = O._ for each n ffi
1, 2 ..... For each intvger n  1, choose a C ® function co n which is
identically equal to 1 on On, has values bctwn 0 and 1, and has support
in On-1. I.vt
(9) v ffi 1  H0.
Then
(10) 'V 1 = '¢-01 = ¢'01 " MI,
where
(11) M 1 = .¢.o -- ¢.o.
In order to-apply 6.$0, wc must first determine to which Sobolcv space the
right-hand side of (I0) belongs.
First, wc claim that
02) o/: ffi of,

248 The Hodge Theorem
from which it follows that ot ¢ C(Oo) and thus belongs to H, for each
s. Now both sides of (12) belong to ome H,, and it follows from 6.18(0
that for (12) to hold, it is sufficient to prove that
(IS) <a, - a,  f, q  >o - 0
for all ? e P. We compute, using (7), (8), 6.18(9), and 6.31(2):
 <,a'19>o- <.f,a,19>o
 <,*a'19>o - 7(*a,9)
- - ffi 0.
Thus (12) holds. Now since L (and hence /') is an operator of order 2,
then M is of order 1; hence M e H_. Thus the right-hand side of (10)
belongs to H_. It follows from the periodic regularity theorem 6.30 that
vte H. Now we let
(14)
Then
(15) .zt ffi /' + M: -- 0 + Mvl,
where the last equality follows from 6.31(4) since M ffi s -- s has
sup in O and  ffi v on O. Now, arguing as fore, we  that the
fit-hand side of (15) lies in He. Thus by 6.30, v e H. Continuing in the
e manner, we obtain
(1 v   with v e H.
Finny, let W,  an on neighrho ofp with W,. O, and let w 
identilly 1 on W, with vues twn 0 and 1, and th supH in O.
en  ffi  for ch n; and so by (16), w  H for eve n. By e
corolla  the Solev lemma 6.22,  psents a C = function u  .
Now if t  C(W,), then
l(t) = = =
= (oa,t)o = (u,t),
and (5) is provod. Expt for a prf of the fact that L is elliptic, this
completes tbe proof of Thrum 6.5.
6.33 Proof of Theorem 6.6 It suffices to show that if m e M, then
there is some neighborhood of m such that if ? is any C ® function on M
with support in that neighborhood, then {?n} has a Cauchy subsequence.
For then we simply cover M by a finite number of such neighborhoods

Reduction to the Periodic Case 29
and take a partition of unity 71 ..... ?x subordinate to this cover. One can
select a subsequence n, such that %n, is Cauchy for each j. Then {t} is
Cauchy for
We reduce the problem to Elidean space by choosing a coordinate
neighborhood ofm with m going top  I-. We continue with the notation
and setup as developed in 6.31 and 6.32; in particular, the norm on M and
the corresponding induced norm on C will now be denoted by II
Let ? be a real-valued C ® function with support in Oe. It suffices to show
that the sequence {(p,} of elements of  has a Cauchy subsequence {?,}
in the 0-norm, since the 0-norm and the/..s-norm agree on C(Oo), and the
L-norm and the norm II I1' are equivalent on C(Oo). According to the
Rellich lemma 6.23, in order to prove that {?n} has a Cauchy subsequence
in the 0-norm, it is sufficient to show that the sequence is bounded in HI.
It follows from the Fundamental Inequality that
(1) I1.11 : const (IIZ:?,II- + I1711-)
 const (IIL?II- + II
: const II ?L.II- + const II(L? - ?L).II- + u,nst II ?11-.
Now
(2)
II qL.ll-x  II ?L.llo ---- II
: mt II qLnll' : const IILII' : const IIAII ".
Let • be a C ® function with values between 0 and 1 which equals 1 on Oo
and has support in V. Then
so that
(3)
Finally,
(4) I1.11-  I1110 = I1.11  mt I111'  nst I111'.
So from (!), (2), (3), and (4) we obtain
But by assumption I111' and II.ll' a unded. Thefore, the quen
{ =} is unded in Hz, and the prf of eom 6.6 is mplete (assuming
the ellipticity of L).

250 The Hodge Theorem
ELLIPTIC1TY OF THE LAPLACE-BELTRAMI OPERATOR
6.4 Remark We shah make use of the following observation. Let
U, V, and W be finite dimensional inner product spaces, and suppose that
v S..L.w
is exact. Let A*: V-- U and B*: W-- V be the adjoints of A and B
respectively. Then B*B + AA* is an isomorphism on V. For let v be a
non-zero element of V. We need only show that (B*B + AA*)v  O. Now
((B*B + AA*)v, v)  (By,By) -F (A*v,A*v).
If Bv  O, then (B*B + AA*)v  O. If Bv  O, then by the exactness,
v lies in the image of A. But A* is injective on the image of A. Thus A*v  O,
which implies that (B*B + AA*)v  O.
We shall apply this to the special case in which .U, V, and W are
A_x(M), A,(M*m), and A+x(M*m) respectively, with the inner products
<o,> = ,(o ^ ,),
and with A and B both left exterior multiplication by   M*.:
(1)
 (M.)  ^,+,(M*.).
According to Exemise 15 of Chapter 2, the sequence (l) is exact; and accord-
ing to {Exercise 14 of Chapter 2, the adjoint of : A,(*)  A+,(M is
(2) (- I)", ,: ^+,(*..)  ^ ,(M*.J.
Thus it follows from the above remarks that
(3) (- l)"**
is an isomorphism on A,(M*).
6.$$ The Laplacian Is Elliptic In order to complete the proofs of
Theorems 6.5 and 6.6, we need to prove that the operator L of 6.32 induced
on Euclidean space by the Laplace-Beltrami operator A via a coordinate
system is elliptic. Proving this, according to 6.28(2), is equivalent to showing
that for each m ¢ M,
(1) A(')(m) , 0
for each smooth form. such that .(m)  0 and for each C ® function ? on
M such that ?(m) ffi 0 but d?(m)  O. Assume that. is a p-form, and let
0  d? ffi   M.*. Recall that
A ffi (--l)-''+'d,d, + (--l)-',d,d.
We compute the left-hand side of (!), keeping in mind at each stage that
q(m) .ffi O. Thus

Exercises 251
Similarly,
d,d,( ?aa)(m) = (d*d( ?a),a)(m) .ffi (2d*?(d?),a)(m)
• ffi (2(d?)*(d?)*a)(m) .ffi 2**(a(m)).
Thus
• d,d(?tg)(m) .ffi 2,,(g(m)).
A(?tg)(m) = --2[(-- 1)",, + (-- l)"-l)**](g(m)),
which is not zero, according to 6.34(3). Hence A is elliptic, and the proof of
the Hodge theorem is at last complete.
6.36 Remark We have seen that for each [ in M*, there is a well-defined
linear transformation .([) on A,(M*,) defined by
(1)
,()(v) ffi A(*)(m),
where v ¢ A,(M*), where a is any p-form such that a(m) ffi v, and where
? is any C ® function such that ?(m) ,ffi 0 and d?(m)  . This linear trans-
formation z() is known as the symbol of the operator A. The ellipticity of
all the operators L on II n obtained from A via coordinate systems is
equivalent to the property that the symbol .() is an isomorphism at each
point m for each non-zero  ¢ M*,. Our computations in 6.35 contain a proof
of the fact that the symbol of the exterior derivative operator d, namely
d(): A(M*) --* A(M*), is simply left exterior multiplication by ; whereas
the symbol of the adjoint  of d is the adjoint of left exterior multiplication
by . In the theory of general partial differential operators on vector bundles
over M, ellipticity is defined, as above, in terms of the symbol.
EXERCISES
Prove that ,A ffi A,.
(a) Prove that the Green's operator G is a bounded linear operator.
(b) Prove that G is self-adjoint on (H) ±.
(c) Prove that G takes bounded sequences into sequences with
Cauchy subsequences.
By using an integral test, prove that the series  (1 + I1) -,
where  ranges over all n-tuples of integers (1 ..... ,), converges
for k  In/2] + 1. (Hint: Induct on n, and evaluate the appro-
priate integrals in terms of spherical coordinates.)
Prove in detail the inequality 6.16(15).
Establish the existence of the matrix A of 6.32(2) together with its
stated properties.

252 he Hodge Theorem
6
g
Derive explicit formulas for d, ,, , and A in Euclidean space.
particular, show that if
o " , o! dxq ^ " " ^ dx,,
then
In
Let ? belong to the C ® periodic functions  on the plane. Prove
that
x y
The Rellich lemma 6.Z says that the natural injection i: Ht-- H,
for s < t is a compact operato.r; that is, it takes bounded sequences
into sequences with convergent subsequences. An analogous example
of this phenomenon is the following. Let C denote the Banach
space of periodic continuous functions on the real line, say with
period 2r, and with norm the sup norm II N ®. Let C  be the subset
of C consisting of functions with continuous first derivative. As
a norm for O we take
Use the Arzeli-Ascoli theorem [27, p. 1261 to prove/hat the natural
injection i: O --* C is a compact operator.
We shall consider a number of elliptic equations of the form Lu  f
on the real line. In each case, f will be smooth and periodic of
period 1, and we look for solutions u also periodic of period 1.
This restriction to periodic functions makes this in essence a problem
on a compact space, the unit circle. We let u' ffi du/dx, etc.
(a) u" ,ffif. This is the simplest example of an elliptic operator
which exhibits all of the essential ingredients of the theory.
What is the formal adjoint of this differential operator'/
Show that there is a solution u (periodic) if and only if f is
orthogonal to the kernel of this adjoint.
(b) u'-u f. What is the kernel (in the periodic functions)
in this case? What are the necessary and sufficient conditions
onffor there to exist a periodic solution?
(c) u" f Show that this operator is formally self-adjoint.
Show that there is a periodic solution if and only if f is orthog-
onal to the kernel; and using the fact that

Exercises 253
show that the unique solution orthogonal to the kernel is
u(x) ,, ft(x -- l)f(t)dt + fx(t-- l)f(t)dt
+ l-2ft(t-- l)f(t)dt.
This explicitly exhibits the Green's operator for this case.
(d) u" + rsu " f. Show that this operator is formally self-adjoint.
What is its kernel ? Derive an explicit formula for the solutioh
u, and show that u is periodic if and only iffis orthogonal to
the kernel.
10 In Theorem 6.11 we used the fact that the Green's operator commutes
with d in proving that each closed form on a compact orientable
Riemannian manifold differs from a harmonic form by an exact
form. Give a proof of this result directly from the Hodge decomposi-
tion theorem without using the Green's operator.
11 A periodic distribution ! is a linear functional on  for which there
exists some integer k  0 and a positive constant c such that
(1) II(q)l  c 
for all q  . Prove that if I is a periodic distribution, then there
is an element u  H_® such that
(2) I() - <u,>0
for all q ¢ . Conversely show that every element of H_.. deter-
mines a periodic distribution via (2).
12 Let  and 0 be n-forms on a compact oriented manifold M" such that
S  " S 0" Prove that  and  differ by an exact form.
13 Show that Theorem 6.6 cannot be strengthened to the assertion of
the existence of a subsequence which is convergent in E'(M).
14 One should observe that in the course of proving Theorem 6.5
we have proved the following regularity theorem. Let L be an
elliptic operator on C ® (the complex m-space valued smooth functions
on ['). Suppose that u is su.Ociently differentiable for Lu to make
sense, and suppose that Lum f where f  C ®. Then u  C".
In fact, we have proved more--every weak solution ! is smooth.
Precisely: Let I be a linear functional on C' which is bounded on
C(V) whenever  is compact, and which satisfies l(L*)
for every   C. Then ! is smooth n the sense that there exists
u  C ® which represents l; that is, l(t)  (u,t) for every t  C.
Such a smooth representative u of a weak solution I is an actual
solution, Lu  f

254 2e Hodge 97teorem
IS
16
Observe that the Cauchy-Riemann operator (i)]i)x)+ i(i)/i)y) is
elliptic. Conclude from the general theory of elliptic operators that
every holomorphic function is C °°. Prove that every holomorphic
function is a complex-valued harmonic function on the plane, aid
use Green's 1st identity (Exercise 5, Chapter 4) to prove that a
holomorphic function with compact support must be identically
zeroo
The Eigenvalueo of the Laphu:ian ]'his is an extended
exercise in which the fundamental properties of the eigenfunctions
and eigenvalues of the Laplacian are developed. Proofs for the more
dicult parts are outlined, and in some cases are given nearly in
full.
Consider the Laplace-Beltrami operator A acting on the p-forms
E'(M) for some fixed p. A real number  corresponding to which
there exists a not identically zero p-form u such that Au   is
called an eigenvalue of A. If Jt is an eigenvalue, then any p-form
u such that Au  tu is called an eigenfunction of A corresponding to
the eigenvalne Jr. The eigenfunctions corresponding to a fixed
form a subspace of E'(M) called the eigenspace of the eigenvalne
(a) Prove that the eigenvalnes of A are non-negative.
(b) Prove that the eigenspaces of A are finite dimensional.
(c) Prove that the eigenvalues have no finite accumulation point.
(d) Prove that eigenfunctions corresponding to distinct eigenvalnes
re orthogonal.
(e) Existce. In order for the above statements to have substance
we must prove that there exist eigenvalues of A. First of all,
zero is an eigenvalue if and only if there m non-trivial har-
monic p-forms on M, and the corresponding eigenspace is
precisely the space/ of harmonic forms. We shall now etab-
lish that A has a positive eigenvalueoin fact, a whole sequence
of eigenvalue$ diverging to + oo. Consider A to be restricted
to (/)±. Then we have A: (H) ± - (/)-, and also w¢ have
the Green's operator G: (H)±-*(/) ±, and
GA   for all  E (/)±. Observe that the eigenvalues
of G I (/)± are the reciprocals of the elgenvalues of A I (/)±"
Let
  sup
Then  > 0 and H¢II < , IIll for every ¢p E (/P')±. We shall
prove that 1/7 is an eigenvalue of A. Let {?}  (/P')± be a
maximizing sequence for 7; that is, 11911  ! and HG9II

(0
Exercises 255
First, we claim that IIO, -- ,11  0, for
I1¢=, -- ?=,11 = -- fl=,ll = -- 2{,, + '
 = ii,ii =.- 2 = ii,ii = +   0.
ond, we claim that I1, - 11  o. For if we let
 -- G -- , then
whe  have u the f that (,}  0. (Why?)
Now e i a ubquen of the , roll i¢ {}, ueh
{} is uchy. De, he a fine funiona] I on (
z()  lira ,} for   '().
Prove that  i a non-tvJa] weak solution of
rom th and the fact that A - ] is elfipfie, conclude that
 = / i an eenvatue of A.
isza¢  @zr igoi. Suppose that we have
va]s s  s  "" •  , and cospondin eienfuncfions
s, s ..... , (ohonoafid) for A  (}. t ,
the subspace of (} snd by {s ..... }. Obese that
 and A map ( @ ,) into it,ll, en de, he
.+s = sup
and proud as in pa (e) ¢o embfish that ,+s = ]+s is an
eienva] of A. Clearly, +s  .
s omlza. t s   .-. be the eienvalues
A on (), whe each eienvalue s included as many times
as the dimension of its eenspace, with a correspondin
ohonormafid qnce of eienfuncfions {}. Let
(). Then
For the prooL let k  the dimension of . Then the exists
k
fl c (H')  such that G --  --  (a,u)u. It follows that

256  Hodge Theem
17
18
for n > k. But, by the definition of .+t,
(h) Uniform Completeness.
E(M) by
<-+IIPU-O as n--oo.
The uniform norm [[0[[® is defined on
flll o = sup (*( ^ *0Xm)) '/*.
From the Sobolev inequality 6.22(!) and the compactness of
M, one can conclude that there exists a large enough integer k
and a constant c > 0 such that
II011®  c I1(1 +
for every 0 c E'(M). Let 0 c EP(M), and let P.(0) 
 (a,u,)u,, where we are continuing with the notation of
part (g). Now Ap.  p.A, so that
iio -- P,,(o)Ii.  c ii(l + A)[o - P.(o)]II
 IIcP - P.,cPII -" o,
where   (1 + A).
We define the operator As: E(M)..E(M) by As(a)- A(A).
Discuss the solvability of AS  .
Consider the operator
acting on Cs(I'). Show that there is no loss of generality in
assuming that a. =. a., and prove that L is elliptic at a point x if
and only if the matrix (ao(x)) is (positive or negative) definite.
In particular, show that the wave equation
Du = =f
is not elliptic, and give an example where
I'u=fcC ° but u¢C ®.

Exercises 257
19 Consider A: E'(M).-* E'(M). Prove that if  is the minimum
eigenvalue of A, and if c > --, then (A + c)a   can be solved
for every   E'(M).
20 The Peter-Weyl Theorem The representative ring of a compact
Lie group G is the ring generated over the complex numbers by
the set of all continuous functionsf for which there is a continuous
homomorphism p: G-,.GI(n,C) for some n such thatf p, for
some choice of i and j. The Peter-Weyl theorem states that the
representative ring i dense in the space of complex-valued continuous
functions on G in the uniform norm. That is, if g is a complex-
valued continuous function on G, and if • > 0 is given, then there
is a function fin the representative ring such that If(o) - g(o)l < •
for all  c G. We outline a proof of this theorem which is based on
the uniform completeness of the e_igenfunctions of the Laplacian.
One can choose a Riemannian structure on G such that each of the
diffeomorphisms !. for  c G (left translation by ) is an isometry
(that is, (v,w), -- (dl.v, dl.w)., for all • c G and all v, w
Since the C ® functions are dense in the space ofcontinuou[ functions
in the uniform norm, and since by Exercise 16(h) the direct sum of
the eigenspaces of the Laplacian is dense in the space of C ® functions
in the uniform norm, it suffices for the Peter-Weyl theorem to prove
that each eigenfunction of the Laplacian A:C®(G)-C®(63
belongs to the representative ring.
Now, G acts on the C ® functions on G by
e(f) ffol. for  c G.
Prove that since the Io are isometrics, this action commutes with
the Laplacian:
A(fo l.) ffi (A f) o I. 63.
lt Vl be the (finite dimensional) eigenspace associated with the
eigenvalue . of A: C®(63- C®(63" Prove that the action of G
leaves V invariant. Then let ?t ..... ?n be a basis of V, and let
Then o -, {g,(o)) is a homomorphism of G --* Gl(n, ,). Prove that
this homomorphism is continuous. Then observe that
so that q belongs to the representative ring.
21 The reader who is familiar with the theory of vector bundles should
observe that the results in this chapter on the Laplace-Beltrami
operator are valid for general elliptic operators on vector bundles.
We outline the statements of the results in this situation:

258 The Hodge Theorem
22
Let E and F be (real or complex) vector bundles over a compact
orientable manifold M. Let C®(E) and C*(F) denote the vector
spaces of smooth sections of E and F. respectively. A (linear)
differential operator I., of order I from E to F is a linear map from
C®(£) to C®(F) which when expressed in terms of local trivializa-
tions of£ and Fyieids an ordinary linear partial differential operator
of order I. The operator/., is elliptic if it is locally elliptic. Equiv-
alently, ellipticity of/., can be defined in terms of the symbol as in
6.36. Observe that for/., to be elliptic, the fibre dimensions of E and
F must be equal. Choose inner products in the fibres £, and F,
which vary smoothly with m, and choose a Riemannian structure
and an orientation on M with respect to which smooth functions
can be integrated over M. By integrating the fibre inner products
over M, we obtain inner products on C®(E) and C*(F). Let
L: C®(E)--* C®(F) be a differential operator. Prove that L has a
formal adjoint L*: C(F)--* C®(E). Assume that L is elliptic.
(a) Prove that ker L and ker L* are finite dimensional.
(b) Prove that C®(E) and C®(F) have the following orthogonal
direct sum decompositions:
C®(F) -- L(C®(E))  ker L*,
C®(E) -- L*(C®(F))  ker L.
Let qn, for n  1, 2, 3 .... , be a periodic C ® function on the plane
which agrees with log iog(l/(r + (l/n))) for 0 : r : t, where
r -- xS]". Show that there is no constant c > 0 such that
Ilqnll® : c IIqllx for all n.
This shows, in the case n -- 2, that the restriction t >. [n/2] + 1
in Corollary (b) of the Sobolev iemma 6.22 is essential.
Let L be a (linear) elliptic operator on the C ® functions on a compact
oriented Riemannian manifold M. Let , be a diffeomorphism of M
which preserves the volume form on M. We say that a C ® functionf
on M is invariant under , iffo , --f, and we say that the operator L
is incariant under , ifLu o ,  L(u o 7') for all C  functions u on M.
Suppose that L and f are invariant under , and thatfis orthogonal
to the kernel of L*. Prove that there is an invariant solution u
of Lu  f.

[1] Bers, L., F..lohn, and M. Schechter. Partial l)ifferentialF, quations.
New York: .lohn Wiley & Sons, Inc., 1964.
[2] Bishop, R. L., and R. J. Crittenden. Geometry of Manifolds. New
York: Academic Press, 1964.
[3] Bredon, (3. E. Sheaf Theory. New York: Mc(3raw-Hill, 1967.
[4] Cartan, H. Sdminaire 19.019.1. Paris: Ecole Normal¢ Suprieure,
1955.
[ Chevalley, C. Theory of Lie Groups I. Princeton, N.J.: Princeton
University Press, 1946.
[6] Fleming, W. H. Functions of Several Variables. Reading, Mass.:
Addison-Wesley, 1965.
-[7] (3odement, R. Topologie Algdbrique et Thdorie des Faisceaux.
Paris: Hermann, 1958.
[8] (3unning, R. C. Lectures on Riemann Surfaces. Princeton, N.J.:
Princeton University Press, 1966.
[9] Helgason, S. Differential Geometry and Symmetric Spaces. New
York: Aeademic Press, 1962.
[10] Hodge, W. V. D. The Theory and Applications of Harmonic Integrals.
2d ed. Cambridge: Cambridge University Press, 1952.
[11] Hurewicz, W. Lectures on Ordinary Differential Fxluations. New
York and Cambridge, Mass.: John Wiley & Sons, Inc., and
Press, 1958.
[12] Jacobson, N. L/e Algebras. New York: John Wiley & Sons, Inc.,
1962.
[13] Kelley, J. L. General Topology. Princeton, N.J.: Van Nostrand
Company, Inc., 1955.
[14] Kervaire, M. A Manifold which does not admit any differen-
tiabie structure. Comment. Math. Helv., 35(1961), 1-14.

ibliography 261
[15] Kobayashi, S., and K. Nomizu. Foundations of Differential Geom-
etry, vol. 1. New York: John Wiley & Sons, Inc., 1963.
[16] Kohn, J.J. Introduccion a )a teoria de integrales harmonicas.
Lecture notes issued by the Centro de Investigacion del IPN,
Mexico, 1963.
[17] Lang, S. Introduction to Differentiable Manifolds. New York:
John Wiley & Sons, Inc., 1962.
[18] Loomis, L. H., and S. Sternberg. Advanced Calculus. Reading,
Mass.: Addison-Wesley, 1968.
[19] Milnor, J. On manifolds homeomorphic to the 7-sphere. Ann.
of Math., 64(1956), 399-405.
[20] .Montgomery, D., and L. Zippin. Topological Transformation
Groups. New York: Interscience, 1955.
[21] Newns, N., and A. Walker. Tangent planes to a differentiable
manifold. 3'. London Math. Soc., 31(1956), 400-407.
[22] Nirenberg, L. On elliptic partial differential equations. Ann. Scuola
Norm. Sup. Pisa, 13(1959), 115-162.
[23] Pontrjagin, L. S. Topological Groups. Princeton, N.J.: Princeton
University Press, 1939.
[24] de Rham, G. Varidt#s Diffcrentiables. Paris: Hermann, 1960. •
[25] Samelson, H. Uher die Sphiren die als Gruppenrfiume auftreten.
Comment Math. Heir., 13(1940), 144-155.
[26] Singer, I. M., and J. Thorpe. Lecture Notes on Elementary Topology
and Geometry. Glenview, Ill. : Scott, Foresman and Company,
1967.
[27] Simmons, G. F. Introduction to Topology md Modern Analysis.
New York:. McGraw-Hill, 1963.
[28] ;panier, E. H. Algebraic Topology. New York: McGraw-Hill
1966.
[29] Spivak, M. Calculus on Manifolds. New York: W. A. Benjamin,
Inc., 1965.
[30] Sternberg, S. Lectures on Differential Geometry. Englewood Cliffs,
N.J. : Prentice-Hall, Inc., 1964.
[31] Woll, J. W., Jr. Functions of Several Variables. New York: Har-
court, Brace & World, Inc., 1966.

19
2O
2O
22
22
26
33
35
36
37
37
41
52
54
55
55
56
56
56
56
56
58
61
62
62
63

(X) 6S
LxY 69
Lx¢o 70
() 73
(wa A.-- A w,,va A..- A v,) 79
• : A(V) A(V) 79, 149, 0
 83
8 , 86
0(n,) 
. 86
x,(a) (a a matx) 86
End(, Aut(V) 87
o(n,C), (n,C) S8
=(X,x.) 98
expx, exp 102
 105
A*, A, A -x 107
sl(n,c), e(n,C) 107
O(N,C), a(n,C) 107
SU(n) eu(n) 108
Sl(n,), eI(n,) 108
o(), (), so() 1
Ad 113
ad 114
/H 20
P", CP" 128
S,(V), Mt(V) 129
&" 141
k', , a 142
L 43
Index of Notation 263
j'D¢o 147
grad f, div V 150
®S,(M,I,), ®H,(M; I,)
K, Z 163
8, 8 163
80 8 163
P(8,U), (8) 163
P  {Sv ;Pv.v} 165
=(s)
#(P) 167
8 @f 169
C* 173
: C*  C  173
Z*(C*), (C*), H*(C*) 173
C*  D* 173
 174
, H*(M,8) . 176
(W* @ 8) 178
80 181
U x, A'(U, 186
d: A'(U, A:(U,K) 186
A*(U,, M,(M,K) 186
*(M, A'(U,G), A(U,G) 187
_s(M,  88
'(M) 189
*(
E*( 191
S,(U), S'(U,, S(U, 192
d: S'(U,K)  Sx(U, 193
S*(U,K), 8"(M,, 8(M,K) 193
8"(U,G) 196
H(M;G), H(M;G) 196
, S(M,G) .... 197

264 Indsx of Nomtlon
2OO
201
2O2
2O7
2O8
212
223
224
227
227
228
228
Iv, I, IIPII
IIPU®
 (-- (,..., .))
., { , ., II II.
K 
L = {L}, L*
C ®, C
c(v)
228
228
243
228
229
231
231
235
236
238
243
244
,]

Action of Lie groups, 112, 121
Adjoint representation, 112 ft.
Ado's theorem, 101
Ane motions, group of, 8
Alexander-Spanier cohomoIo, 186 ft.
with supports, 215
Alternating multilinear maps, 56 ft.
Anti-derivation, 61
Axiomatic sheaf cohomology, 176 ft.
multiplicative structure, 207 ft.
with supports, 214 ft.
Ball in Euclidean space, 4
Bilinear forms, automorphisms and
derivations of, 118-119
Bilinear operations, automorphisms and
derivations of, I 17-II $
Boundary
differentiable singular, 154
of a -simplex, 142
Canan lemma, 80 (£x. 16)
Cartesian product of
manifolds, 7, 52 (£x. 24)
mappings, 3
SetS, 
Cauchy-Riemann operator, 254 (Ex. 15)
Cech cohomoloKy, 00 ft.
with supports, 215
Centered coordinate system, 5
Chain, 142
Chain complex, 199
Chain rule, 17
Class C', 5-6
Class C 
differentiable manifold, 6
function on Euclidean space, 5
mapping, S
Classical cohomology theories. ,See
Cohomology theories.
Cochain complex, 173 ft. •
Cochain complex (cost)
coboundaries, 173
coboundary operator, 173
cocycles, 173
cohomoloy of, 173
tensor product of, 207
Cochain map, 173
Coframe field, 149
Cohomoloy with coefficients in sheaves,
176 ft.
Cohomoloy theories
Alexander-Spanier, 186 ft., 215
ch, 200 ft., 215
differentiable singular, 191 ft., 205-207,
212-215, 227
de Rham, 153 ft., 189 ft., 205-207,
211-215, 217 (Ex. 21), 225-227
singular, 191 ft., 205, 212-215, 217
(Ex. 19)
Commutative diagram, 3
Completely interable distribution, 42
Complex analytic structure, 6
Complex general linear group, 88
connectedness, 136 (Ex. 25)
exponential map, 105 ft., 134 (Exs. 14,
15), 135 (£x. 22)
polar decomposition, 136 (Ex. 24)
subgroups, 107-108
Complex manifold, 6
Complex n-space, 4, 7
Complex number field, 4, 83
Complex orthogonal group, 107-108
Complex proje_tive space, 128
Composition of mappings, 3
Constant sheaf, 164
Coordinate functions on Euclidean space,4
Coordinate function-% 5
Coordinate map, 5
Coordinate system, 5
centered, 5
cubic, 5
slice of, 27

266 Foundations of Differentiabl¢ Manifolds and Lie Groups
Cotangent bundle, 19
Countability, second itxiom of, 6, 8 ft.
Cover, 8
locally finite, 8
open, $
refinermt, $
subcover, 8
Covering spaces, 98 ft., 132 (£x.
base, 98
covering, 98
evenly covered set, 98
from Lie group homomorphism, 100
Cube in Euclidean space, 4
Cubic coordinate system, 5
Curve, 17
integral, 36 ft.
piecewise smooth, 34
smooth on[a,b], 34
tangent veclor to, 17
Cycle, differentiable singular, 154
de Rham. See ubr R-listing.
Derivation
on functions, 12
on forms, 61
Derivative, 4
exterior, 65 ft.
partial, 4
Determinants, 79 (Ex. 12)
Difteomorphism, 22-23
Difference quotients, 235 ft.
Differentiable (see also Class Ce), 6
Differentiable manifold (see a/so
Manifold), $ ft.
Diffen.-ntiable singular cohomology,
191 ft., 205-207, 227
multiplicative structure, 212-214
with supports, 215
Difterentiable structure, 5-6
of class &, 5
of class C , 5
of class C,*, 6
complex analytic, 6
Differential
of a function, 16-17
of coordinate functions, 17
higher order, ;'0-22
of a map, 16
Difterential forms, 62 ft.
closed, 153
differential ideals, 74
exact, 153
exterior derivative, 65
independent 1-forms, 73
interior multiplication by vector fields,
left invariant, 88 ft.
Lie derivative, 70
eftect of mappings, 68
Differential fonm (eot)
Maurer-Cartan forms, 88
periods, 155
de Rham cohomology, 153 ft., 189 ft..
205 ft.. 211,215, 226
Difterential ideal, 74
integral manifold, 74-75
maxinml integral manifold, 74-75
Direct limit of/'-moduies. 167
Distributions, 41 ft.
annihilating forms, 73
annihilating ideal, 73
Frobonius theorem, 42, 48, 75
integral manifold, 42
involutive (completely integreble), 42,
74
maximal integral manifold, 48, 75
Divergence of a vector field. 150
Divergence theorem, 151, 157 (Ex. 4)
Effective action of a Lie group, 12.]
Elliptic equations, simple examples, 252
(e'x. 9)
Elliptic operators, 240 ft.
fundamental inequality, 240
on vector bundles, 257 (Ex. 21)
invariant solutions, 258 (Ex. 23)
Equivalence of submanifolds, 26
Euclidean space, 4
integration on. 140-141
as a Lie ..up, 83
as a manifold, 6
de Rham cA)homology of, 156, 158
(Ex. 10)
standard orientation, 140
Exact sequences. 165
Exponential map, 102 ft., 116
for general linear group, 105 ft., 134
(Exs. 10, 14, 15), 135(Ex. 22)
relation between subgroup and
subalgebra, 104
Extension of vector field, 39-40
Exterior algebra, 56 ft.
anti-derivations, 61
derivations, 61
dualities, 58 ft.
endomorphisms of degree k, 61
inner products on, 79 (Ex. 13)
interior product. 61
star operator. 79 (Ex. 13)
universal mapping property, 57-58
Exterior k-bundle, 62
Exterior algebra bundle, 62
Exterior derivative, 65 ft.
Exterior (wedge) product, 56, 60
Family of supports, 214
Fine resolution, 178
Formal adjoint, 239
Forms. See Differential forms.

Pourer szrks. 22 ft.
Fram field, 149
Frobznius t, 42, , 75
I, 4
Fui, 3
dte, 4. S
wmt, 4
t, 23
FdtE up, M
F ty, 
Ft   kul.
143
 linear group, 7, 83, S6-S8
components, 131
exponential map, 10 ft.. 14 (£x. 10)
polar decomposition, 11, 136 (Ex.
subgroups, 107-108
Set also Complgx gemal linear group.
Germ of a function, 12
Gradient. 150
Graph of a map, 75
as integral nnifold, 75 ft.
Grassmann manifold, 130
Green's identities, 155 (Ex. $)
Green's operato, 225
Harmonic function, 222
Harmonic form, 223
Hilbert's Fifth Conjecture, 110
Hodge decomposition theorem, 223
Homogeneous manifolds, 120 ft., 125 ft.
Homology
real continuous singular, 227
real differcntiable singular, 154, 206
Homomorphism
Lie group, 89
of sheaf cohomology theories, 180 ft.
Homotopy operator, 157, 190, 194. 199,
202
Identity map, 3
Imbedding, 22
Immersion, 22
Implicit function theorems. 30 ft.
classical, 30
Independent set of functions, 23
lnjective mapping, 3
Integral curve, 36ff.
Integral manifold, 42, 74
maximal, 48, 74
Integration, 140 ft.
over chains, 141 ft.
in Euclidean space, 140
of n-forms in Euclidean space, 141
on Lie groups, 151 ft.
on oriented manifolds, 145 ft.
on Riemannian manifolds, 1,9 ft.
Index 267
Interior multiplication, 61
by vcto field 6S
lnvariant diffential opato, 255 (Ex.
23)
Invariant oluticm, 255 (Ex. 23)
Inverse function thmrn, 23
Involutiv® di'ibutiom, 42, 74
lsotropy group, 1
Jacob/identity, 6, 84
Jacobian, 17
K-modules, 163
direct limit, 167
totionless, 172, 207
Kronccker index, 4
L, norm, 22S, 243
Laplace-Beltrami operator, 220 ft.
eigenvalues, 2.54 (£x. 16)
ellipticity, 2-1
in Euclidsan spacz, 158 (x. ,
(x. )
symbol, 251
ltlaian, 158 (£x. b'), 220 ft.
invariant forms, 88-89, 90-92
Left invariant vecto fields, S4 ft.
completeness, 103
Left multiplication, 61
Left translation, S4
Lie algebra, 36, S4
abelian, S4
center, 116
homomorphism, 90
isomorphism, 90
of a Lie group. $6
of n × n matrices, S4, $6-85. 107-108
LlUbkbbrtation, 90
a, 93
et, 36, 75 (£x. 6), S4
as Lie derivative, 70
Lie derivative
of differential forms, 70
of vtor fiedds, 69
Lie group. $2 ft.
abelian, 116, 135 (Ex. 15)
action on manifold 112-113, 123
adjoint rgprentation, 112 ft.
associated with each Lie algebra, 101,
117
automorphism, $9
automorphism group, 119, 135 (Ex. 20)
bi-invariant metric, 155 (Ex. 9)
center, 116
closed subgroup, 93, 110 ft., 135 (Ex.
1)

268 Foundations of Differentiable Manifolds and Lie Groups
Lie 
cled-u .lFep_ toponogy,
cotmivity, 130
tin h,
p, 83, 8, 107-1
t p, 102
 ti, 101
f , IM {. 12)
 a h fold, 1

h th ly
, 101
iti t,
itity t, 82
intion on,
 mvat f, 88
t int or i
  of, 86
 sunup
 sumup, 115
l-ter sunup. 102
ota,
ght int or  1 (.
t tti,
mply ned. 101
dmply  ng gup,
101
tru tts.
sunup (s L oup)
imular, 152
Lie suroup, 92 ft.
ci, 93, 110 ft.,
ci sunup tl, 97
ulval of, 93, 997
nol, 11
l-mer, 102
relation t
sulbs & t Lie alum,
q of, 93, 9
ting, 34
r diffti tor, 8
Lr
i l-er p  a or
ld, 39
Lay Ein s,
with  difffia
th -diffohk u,
Mol 5 ft.
m, 1 ft.
Manifolds ()
int.ep, of a diatribution or differential
noex, 42, 74
inte[Fation on, 140 ft.
nmxmal integral, 48, 74
metrizability, 8
normality, 8
open submanifold, 7
orientabie, 138
product manifold, 7, $2 (Ex. 24)
Riemanninn, $2 (Ex. 23), 149-151,
22O ft.
second axiom of countability, 6, 8 ft.
slices, 27
submaaifold, 22, 26-27
Manifold structure, 6
Mappings
commutative diagram, 3
composition, 3
diffemolphism, 22
factoring through submanifolc:k, 25-26,
47
function, 3
identity, 3
imbedding, 22
immersion, 22
non-singular, 16
one-to-one (injective), 3
onto (surjective), 3
restriction, 3
submaolfold, 22
Matrices, 7, 83, 84, 86-88, 107-108,
12-132, 136 (Ex. 25)
exponential nmp, 105 ft., 134 (F.xs. 10,
14, 1S), 13S (Ex. 22)
polar decomposition, 131, 136 (Exs.
23, 24)
positive definite, 131
Maurer-Cartan fonm, 88
Maximal integral manifold, 48, 74
Modular function, 152
Multi-index notation, $, 227
Neighborhood, 4
Non-singular mapping, 16
Non-singular pairing, $8
Norms
L, 228, 243
Sobolev, 231
-parameter group of a vector field. 39
-parameter subgroup, 102
Orientation. 138 ft.
vector s]e, 79 (x. 13)
rthogonal group 33, 108, 119, 125-131
components, 130

Index 269
for manifolds, 9
Prti! derivfiv 4
Partition of unity
  8
For , 170
sup of, 
i tu 3 (. 11)
s ofFo 1, 217 (. 21)
Poi mio 131, 136 (. ,
 Wt, 169
Quotient shtf, 165
Real continuous singular homology, 227
Real difftiahle singular homology,
154, 206
Real projectivz space, 128
orientation, 140, 157 (Ex. 2)
Real special liner group, 10
Regular domain, 145 ft.
Regularity theorem, 223, 242, 245 ft.,
253 (Ex. 14)
Rellich lemnm, 237
Representations of Lie groups, g9
orthogonal and unitary, 152-153
Resolution. 178
de Rham cohomology, 153 ft., 189 ft.,
205-207, 217 (Ex. 21), 22--227
of Euclidean space, 156, 155 (Ex. 10)
other examples, 158 (Exs. 10, 11), 159
(Exs. 16, 17), 160 (Exs. 15, 19)
multiplicative structure, 211
supports, 215
of the unit circle, 153
de Rlmm thcorzm, 154-155, 205 ft., 214.
215
Rlerrmnnian manifold, 52 (Ex. 23), 220 IT.
integration on, 149 IT.
volume, 149
volume form. 149, 155 (Ex. 6), 160
(Ex. 20)
ittemannitn structure, S2 (z. 23)
Right invariant vector field, 135 (F.x. 16)
Right transhttiotl, g4
Schwartz inequity, 232
Second axiom of countability, 6, 8 ft.
Semi-locally l-connected, 98
Set notation, 2-4
sgu of a permutation, 6
Sheaf, 162 ft.
assoctl prmim', 166ff.
constant, 164
of" gzrms of" discontinuous sctiom, 181
homomor.phism, 164
of" g-modules, 15
parttfion of tmity,
projcgtion, 163
quotimtt, 165
resolution, 178
subshf, 164
stalk, 163
tensor product, 169
tovsionless, 172
Sheaf cohomolog theory, 176 ft.
existence, 178 ft.
homomotism of, 180 ft.
multiplicativz structure, 207 ft.
with supports, 215
uniqueness, 181 ft.
Simplex, 141,191
boundary of', 142
continuous singular, 191
differentiahle singular, 141
face of', 142
oriented, 146
regular, 146
standard, 141
Simply connected space, 98
Lie group, 99, 101
Singular cohomology, 191 ft., 205, 217
(£x. 19)
multiplicativz structure, 212
with supports, 215
Skew-Hermitian matrices, 107-10
Skew line on torus, 51 (Ex. 21)
Skew-symmetric matrices, 107-10
Skyscraper sheaf, 171
Slices, 27
Smooth (class C'm), 6
Sobolev lemma, 237, 238 (£x. 22)
Sobolev spaces, 231 IT.
Special linear group, 107-10 ....

270 Foundatio-.  Dertntlable Malfoid and I Oroups
Sphere, 7, 33
as h manifold, 1 ft.
orkntation, 10
Sta operator, 7 (a. 13), 0 ft. ".
Stiefel numifoid,
Stok' theorem, 14, 14, 151
Submanifolds, 22
Tangent bundle, 19-20
Tangeot spa 12
Tangent vctog, 11 ft.
from coordinate system, 14--1
to curvm, 17
higher order, 20-22
Temors. 55 ft.
homogatus, $6
type (r,0, 55
Tensor algebra. 55 ft.
dit,  ft.
Tr k of t (r 62
Tr !,  ft.
 n p, 
 fful, 169
of , 169
of rmolutio
m.ppiN ipty, ss
Topoog group, tO
Togaionless resolution, 175
Torus, 1 (Ex. 21), , 13 (. 18)
Trace 0 matrices,
Transitive actio 123
Unity group, 10-l(l, I0, I
21)
of xtzrior al
Vector fields,
complete, 39
along a curve, 34
loca C  .emion, 39-40
acting on functions, 35
integral ¢urv 6 ft.
as liftings to T(M), 4
left invarnt, 4 ft.
Lie derivative, 69 ft.
along a mpping 39
as a manifold, 7, 6
oriontatimt, 9 (Ex. 13)
Volun 19
Volume form, la9, 1 (Ex. 6), 160 (Ex.
2O)
Wave equation, 256 (K. 1.8)
Weak solution, 222
w. (xtor) podt, S6, 60