INTRODUCTION TO
SMOOTH MANIFOLDS
by John M. Lee
University of Washington
Department of Mathematics
www.pdfgrip.com
Corrections to
Introduction to Smooth Manifolds
Version 3.0
by John M. Lee
April 18, 2001
• Page 4, second paragraph after Lemma 1.1: Omit redundant “the.”
• Page 11, Example 1.6: In the third line above the second equation, change “for each j”
to “for each i.”
• Page 12, Example 1.7, line 5: Change “manifold” to “smooth manifold.”
± −1
• Page 13, Example 1.11: Just before and in the displayed equation, change ϕ±
to
j ◦ (ϕi )
±
± −1
ϕi ◦ (ϕj ) (twice).
• Page 21, Problem 1-3: Change the definition of σ to σ(x) = −σ(−x). (This is stereographic
projection from the south pole.)
• Page 24, 5th line below the heading: “multiples” is misspelled.
• Page 24, last paragraph before Exercise 2.1: There is a subtle problem with the definition of smooth maps between manifolds given here, because this definition doesn’t obviously
imply that smooth maps are continuous. Here’s how to fix it. Replace the third sentence of
this paragraph by “We say F is a smooth map if for any p ∈ M , there exist charts (U, ϕ) containing p and (V, ψ) containing F (p) such that F (U ) ⊂ V and the composite map ψ ◦ F ◦ ϕ−1
is smooth from ϕ(U ) to ψ(V ). Note that this definition implies, in particular, that every
smooth map is continuous: If W ⊂ N is any open set, for each p ∈ F −1 (W ) we can choose a
coordinate domain V ⊂ W containing F (p), and then the definition guarantees the existence
of a coordinate domain U containing p such that U ⊂ F −1 (V ) ⊂ F −1 (W ), which implies that
F −1 (W ) is open.”
• Page 25, Lemma 2.2: Change the statement of this lemma to “Let M , N be smooth
manifolds and let F : M → N be any map. Show that F is smooth if and only if it is
continuous and satisfies the following condition: Given any smooth atlases {(Uα , ϕα )} and
{(Vβ , ψβ )} for M and N , respectively, each map ψβ ◦ F ◦ ϕ−1
α is smooth on its domain of
definition.”
• Page 30, line 6: Change “topology of M ” to “topology of M .”
• Page 31, Example 2.10(e), first line: Change “complex” to “real.”
• Page 36, Exercise 2.9: Replace the first sentence of the exercise by the following: “Show
that a cover {Uα } of X by precompact open sets is locally finite if and only if each Uα
intersects Uβ for only finitely many β.”
• Page 39, line 5: Insert a period after the word “manifold.”
• Page 40, Problem 2-2: The first sentence should read “Let M = Bn , . . . .”
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• Page 41, line 5 from bottom: Change “abstract definition of” to “abstract definition of
tangent vectors.”
• Page 41, line 4 from bottom: “showing” is misspelled.
• Page 43, line 4: Change Sn to Sn−1 (twice).
• Page 48, last displayed equation: The derivative should be evaluated at t = 0:
va f =
d
dt
f (a + tv).
t=0
• Page 51, line 15: Insert “p ↔ p,” before the word “and.”
• Page 54, two lines below the first displayed equation: Insert “it” before “is customary.”
• Page 57, four lines below the first displayed equation: Delete “depending on context.”
• Page 58, line 5: Change (2.2) to (3.6).
• Page 59, just below the commutative diagram: Replace the first phrase after the
diagram by “and for each q ∈ U , the restriction of Φ to Eq is a linear isomorphism from Eq
to {q} ì Rk
= Rk .
ã Page 60, Exercise 3.6: Move this exercise after the second paragraph on this page.
• Page 60, last sentence before the heading “Vector Fields”: Change “3.13” to “Lemma
3.12.”
• Page 63, Lemma 3.17: Both vector fields are mistakenly written as X in several places in
this lemma and its proof. In fact, to be consistent with the surrounding text, they should
have been called Y and Z. Replace the entire lemma and proof by:
Lemma 3.17. Suppose F : M → N is a smooth map, Y ∈ T(M ), and Z ∈ T(N ). Then Y
and Z are F -related if and only if for every smooth function f defined on an open subset of
N,
Y (f ◦ F ) = (Zf ) ◦ F.
(3.7)
Proof. For any p ∈ M ,
Y (f ◦ F )(p) = Yp (f ◦ F )
= (F∗ Yp )f,
while
(Zf ) ◦ F (p) = (Zf )(F (p))
= ZF (p) f.
Thus (3.7) is true for all f if and only if F∗ Yp = ZF (p) for all p, i.e., if and only if Y and Z
are F -related.
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• Page 64, Problem 3-2: The third displayed equation should be
α−1 (X1 , . . . , Xk ) = j1∗ X1 + · · · + jk∗ Xk .
• Page 74, second line from bottom: Delete the symbol γ∗ .
• Page 83, last displayed equation: Should be changed to
f (q) − f (q) =
q
p0
ω−
q
p1
ω=
p1
ω.
p0
• Page 86, Example 4.26, line 1: Change “Example 4.7” to “Example 4.18.”
• Page 95, part (f ): Change “(y − 2)2 + z 2 + 1” to “(y − 2)2 + z 2 = 1.”
• Page 111, second line from bottom: Change “W is open” to “π(W ) is open.”
• Page 112, 5th line from bottom: Change q ∈ M to q ∈ N .
• Page 117, second line under (5.10): Change “observe that E has rank k . . . ” to “observe
that E has rank less than or equal to k . . . .”
• Page 119, fourth line under the heading “Immersed Submanifolds”: change the
word “groups” to “subgroups.”
• Page 120, 5th line after the subheading: Insert missing right parenthesis after “topology.”
• Page 121, line 7 from bottom: Change “fo” to “for.”
• Page 126, Problem 5-3: Delete this problem. (The answer is already given in Example
5.2.)
• Page 127, Problem 5-11: Change the definition of S to
S = {(x, y) : |x| = 1 and |y| ≤ 1, or |y| = 1 and |x| ≤ 1}.
• Page 127, Problem 5-14: Delete part (b).
• Page 129, line 4 from bottom: Change “in the sense . . . ” to “in a sense . . . .”
• Page 130, second line from bottom: Change F (Bj ) to F (A ∩ B ∩ Bj ).
• Page 133, proof of Theorem 6.9: In the second paragraph of the proof, replace the first
sentence by “For each i, let ϕi ∈ C ∞ (M ) be a bump function that is supported in Wi and
identically equal to 1 on U i .”
• Page 139, last displayed equation: Change
vj
∂
∂xj
to
vj
∂
∂xj
.
x
• Page 142, three lines above the last displayed equation: Change “a δ-approximation”
to “δ-close.”
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• Page 143, proof of Proposition 6.20, first line: Change “H : M × I → M ” to “H : M ì
I N .
ã Page 148, line 2: Change “by continuity” to “by continuity of πG ◦ Θ−1 .”
• Page 154, paragraph 2, line 2: Change “contained in GK = {g ∈ G : (g · K) ∩ K = ∅}”
to “contained in GK = {g ∈ G : (g · K ) ∩ K = ∅}, where K = K ∪ {p}, . . . .”
• Page 154, paragraph below conditions (i) and (ii): Change U to W (twice).
• Page 167, Problem 7-7(c): Add the hypothesis that n > 1.
• Page 169, Problem 7-24: Change U(n) to U(n + 1).
• Page 176, proof of Proposition 8.3, lines 1, 2, and 11: Change X to Y (three times).
• Page 178, first full paragraph: Add the following sentence at the end of this paragraph:
“Applying this observation to V = (V ∗ )∗ and W = (W ∗ )∗ proves (b).
• Page 183, third display: In the second line, change T στ to T τ σ .
• Page 183, first line after the third display: Change “η = στ ” to “η = τ σ.”
• Page 191, Corollary 8.20: This corollary, and the paragraph preceding it, should be moved
to page 195, immediately following the proof of Proposition 8.26.
• Page 192, first display: Change dt to dϕ (three times).
• Page 204, Exercise 9.1(d): Change “independent” to “dependent.”
• Page 206, last displayed equation: Change e123 (X, Y, X) to e123 (X, Y, Z).
• Page 220, Exercise 9.7: Change the statement to “Let (V, ω) be a 2n-dimensional symplectic vector space, . . . .”
• Page 222, line 6 from bottom: Change “pullback” to “dual map.”
∗
• Page 222, line 5 from bottom: Change T(p,η) (T ∗ M ) to T(p,η)
(T ∗ M ).
• Page 225, Problem 9-1: In the last line, change det(v1 , . . . , vn ) to | det(v1 , . . . , vn )|.
• Page 225, Problem 9-6(a): Change the definition of the coordinates to “(x, y, z) =
(ρ sin ϕ cos θ, ρ sin ϕ sin θ, ρ cos ϕ)” [insert missing factors of ρ].
• Page 227, Problem 9-9: Replace the first two sentences of the problem by the following:
“Let (V, ω) be a symplectic vector space of dimension 2n. Show that for every symplectic,
isotropic, coisotropic, or Lagrangian subspace S ⊂ V , there exists a symplectic basis (Ai , Bi )
for V with the following property:”
• Page 229, 9th line from bottom: Delete the redundant “which.”
• Page 235, 3rd line from bottom: Delete the word “locally.”
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• Page 239, second display: Two occurrences of dxi should be changed to dx1 , so the
equation reads:
n
(N Ω)|∂M = f
(−1)i−1 dxi (N )dx1 |∂M ∧ · · · ∧ dxi |∂M ∧ · · · ∧ dxn |∂M
i=1
= (−1)n−1 f dxn (N )dx1 |∂M ∧ · · · ∧ dxn−1 |∂M .
• Page 239, third line from bottom: Change Rn to Rn−1 .
• Page 249, equation (10.6): Change ωi to ωn (twice).
• Page 251, third displayed equation: Should be changed to
γ ∗ df =
df =
γ
[a,b]
M
df = f (γ(b)) − f (γ(b)).
• Page 256, equation (10.10): Change σ1 and σ0 to σ1 and σ0 , respectively.
• Pages 259–267: Change every occurrence of ·, · to ·, · g .
• Page 261, proof of Lemma 10.38, 7th line: Change “Corollary 10.40” to “Proposition
10.37.”
• Page 261, fourth display and the two sentences following it: Change M to S (four
times).
• Page 262, second line from bottom: Change “domain with smooth boundary” to “regular
domain.”
• Page 263, second paragraph after the subheading: Add the following sentence at the
end of the paragraph: “Since β takes smooth sections to smooth sections, it also defines an
isomorphism (which we denote by the same symbol) β : T(M ) → A2 (M ).”
• Page 267, Problem 10-16: In parts (b) and (c), change “connected” to “compact and
connected.”
• Page 267, Problem 10-17: Add the phrase “(without boundary)” after “Riemannian
manifold.”
• Page 271, line 5: Change “Example 4.23” to Example 4.26,” and change “closed 1-form”
to “1-form.”
• Page 274, line above equation (11.3): Interchange M and N .
• Page 275, two lines above Case I: Change “can be written as . . . ” to “can be written
locally as . . . .”
• Page 275, Case I: In the first line, delete the phrase “because dt ∧ dt = 0.”
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• Page 275, line above the last display: Change this line to “On the other hand, because
dt ∧ dt = 0,”.
• Page 280, proof of Theorem 11.15, fourth line: Change I[aω] to I[aΩ].
• Page 281, proof of Theorem 11.18, third line: Change “α : M → M ” to “α : M → M .”
• Page 284, line 5: Change y < R to y < −R, and change F to E.
• Page 289, line above equation (11.11): Change Ap (V ) to Ap (U ).
• Page 293, line 7: Change σ ◦ F to F ◦ σ.
• Page 295, equation (11.18): Change δ to ∂ ∗ (twice).
• Page 296, third line below the subheading: Change “p-form on M ” to “p-form ω on
M .”
• Page 298, second line after equation (11.19): Change “(p−1)-chain” to “(p+1)-chain.”
• Page 299, proof of Lemma 11.32, last line: Change this sentence to “This implies
I(F ∗ [ω])[σ] = I[ω][F ◦ σ] = I[ω](F∗ [σ]) = F ∗ (I[ω])[σ], which was to be proved.”
• Page 299, proof of Lemma 11.33, fifth line: Change “(p − 1)-form” to “p-form,” and
change “p-chain” to “smooth (p − 1)-chain.”
• Page 299, line 3 from bottom: Change the first Ap (U ) to Ap−1 (U ), and change the second
to Ap−1 (V ).
• Page 299, line 2 from bottom: Change “smooth simplices” to “smooth chains.”
• Page 300, proof of Theorem 11.34, Step 1: In the second line, change 11.27(c) to
11.27(b).
• Page 300, last line: Change “spanned” to “generated.” Also, change “0-simplex” to “singular 0-simplex.”
• Page 302, last line before Step 5: Replace the last sentence by “Finally, U ∩ V is de
Rham because it is the disjoint union of the sets Bm ∩ Bm+1 , each of which has a finite de
Rham cover consisting of sets of the form Uα ∩ Uβ , where Uα and Uβ are basis sets used to
define Bm and Bm+1 , respectively. Thus U ∪ V is de Rham by Step 3.”
• Page 303, Problem 11-2(b): In the displayed equation, change Pi to Pi (ω).
• Page 304, Problem 11-4, line 4: Change “A smooth submanifold” to “A smooth oriented
submanifold.”
• Page 304, Problem 11-4, line 6: Assume S ⊂ M is compact.
• Page 304, Problem 11-4, line 9: Change 1985 to 1982.
• Page 309, fifth line after the first display: Change “This the reason” to “This is the
reason . . . .”
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• Page 313, two lines above Lemma 12.7: Change “local” to “global.”
• Page 313, second display: Add the following condition before the two that are already
listed:
If s ∈ Dp , then Dθ(s,p) = {t ∈ R : s + t ∈ Dp }.
• Page 329, proof of Lemma 13.1: Replace the first three sentences of the proof by the
following: “First we prove that [V, W ]p is a tangent vector, i.e., a linear derivation of C ∞ (M )
at p. It is obviously linear over R, so only the product rule needs to be checked.”
• Page 336, proof of Proposition 13.9, second line: Change (−t, p) to (t, p).
• Page 338, line below the first dispayed equation: Change W to S.
• Page 338, second and fourth displayed equations: Change (0, (0, . . . , 0, xk+1 , . . . , xn ))
to (0, . . . , 0, xk+1 , . . . , xn ) (three times).
• Page 338, fourth displayed equation: Make the following substitution (four times):
∂
∂
→
i
∂x
∂xi
.
x0
• Page 338, second line after the fourth displayed equation: Change (Vi )ψ(x) f to
(Vi )ψ(x0 ) f .
• Page 339, Proposition 13.11(e): Change η to ω.
• Page 340, last displayed equation: Change df (y) to df (Y ) in the first line.
• Page 341, proof of Proposition 13.14: In the second paragraph of the proof, change
“Proposition 13.11(d)” to “Proposition 13.11(b).”
• Page 342, last displayed equation: In the second line of the display, change Y Tij dxi to
Y Tij dxi ⊗ dxj .
• Page 343, proof of Lemma 13.17, last line: Change (LV W ) to (LX τ ).
• Page 343, proof of Proposition 13.18, first line: Change “θt∗ τ = τ for all t” to “θt∗ τ = τ
on the domain of θt for each t.”
• Page 351, displayed equations: Change H to X (three times).
• Page 360, sentence before Proposition 14.6: Change “consequences” to “consequence,”
and change “lemma” to “proposition.”
• Page 365, Example 14.11(g): Change “z-axis” to “y-axis.”
• Page 367, line 3: Delete the parenthetical remark.
• Page 367, subheading after the proof of Lemma 14.14: Frobenius should be capitalized.
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• Page 374, second display: In the last term of the equation, change Vhg to Vhg .
• Page 376, line 9 from bottom: Add missing right parenthesis in Lie(GL(n, R)).
• Page 388, last line: Delete “⊂ gl(n, R).”
• Page 394, line 3: Change (exp Yi )n to (exp Yi )ni .
• Page 426, statement of Theorem A.20: Change the first sentence to “ . . . such that the
partial derivatives ∂f /∂xi : U × [a, b] → R are also continuous for i = 1, . . . , n.”
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John M. Lee
Introduction to
Smooth Manifolds
Version 3.0
December 31, 2000
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iv
John M. Lee
University of Washington
Department of Mathematics
Seattle, WA 98195-4350
USA
/>
c 2000 by John M. Lee
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Preface
This book is an introductory graduate-level textbook on the theory of
smooth manifolds, for students who already have a solid acquaintance with
general topology, the fundamental group, and covering spaces, as well as
basic undergraduate linear algebra and real analysis. It is a natural sequel
to my earlier book on topological manifolds [Lee00].
This subject is often called “differential geometry.” I have mostly avoided
this term, however, because it applies more properly to the study of smooth
manifolds endowed with some extra structure, such as a Riemannian metric, a symplectic structure, a Lie group structure, or a foliation, and of the
properties that are invariant under maps that preserve the structure. Although I do treat all of these subjects in this book, they are treated more as
interesting examples to which to apply the general theory than as objects
of study in their own right. A student who finishes this book should be
well prepared to go on to study any of these specialized subjects in much
greater depth.
The book is organized roughly as follows. Chapters 1 through 4 are
mainly definitions. It is the bane of this subject that there are so many
definitions that must be piled on top of one another before anything interesting can be said, much less proved. I have tried, nonetheless, to bring
in significant applications as early and as often as possible. The first one
comes at the end of Chapter 4, where I show how to generalize the classical
theory of line integrals to manifolds.
The next three chapters, 5 through 7, present the first of four major
foundational theorems on which all of smooth manifolds theory rests—the
inverse function theorem—and some applications of it: to submanifold the-
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vi
Preface
ory, embeddings of smooth manifolds into Euclidean spaces, approximation
of continuous maps by smooth ones, and quotients of manifolds by group
actions.
The next four chapters, 8 through 11, focus on tensors and tensor fields
on manifolds, and progress from Riemannian metrics through differential
forms, integration, and Stokes’s theorem (the second of the four foundational theorems), culminating in the de Rham theorem, which relates differential forms on a smooth manifold to its topology via its singular cohomology groups. The proof of the de Rham theorem I give is an adaptation
of the beautiful and elementary argument discovered in 1962 by Glen E.
Bredon [Bre93].
The last group of four chapters, 12 through 15, explores the circle of
ideas surrounding integral curves and flows of vector fields, which are the
smooth-manifold version of systems of ordinary differential equations. I
prove a basic version of the existence, uniqueness, and smoothness theorem for ordinary differential equations in Chapter 12, and use that to prove
the fundamental theorem on flows, the third foundational theorem. After
a technical excursion into the theory of Lie derivatives, flows are applied
to study foliations and the Frobenius theorem (the last of the four foundational theorems), and to explore the relationship between Lie groups and
Lie algebras.
The Appendix (which most readers should read first, or at least skim)
contains a very cursory summary of prerequisite material on linear algebra
and calculus that is used throughout the book. One large piece of prerequisite material that should probably be in the Appendix, but is not yet,
is a summary of general topology, including the theory of the fundamental
group and covering spaces. If you need a review of that, you will have to
look at another book. (Of course, I recommend [Lee00], but there are many
other texts that will serve at least as well!)
This is still a work in progress, and there are bound to be errors and
omissions. Thus you will have to be particularly alert for typos and other
mistakes. Please let me know as soon as possible when you find any errors,
unclear descriptions, or questionable statements. I’ll post corrections on
the Web for anything that is wrong or misleading.
I apologize in advance for the dearth of illustrations. I plan eventually
to include copious drawings in the book, but I have not yet had time to
generate them. Any instructor teaching from this book should be sure to
draw all the relevant pictures in class, and any student studying from them
should make an effort to draw pictures whenever possible.
Acknowledgments. There are many people who have contributed to the development of this book in indispensable ways. I would like to mention especially Judith Arms and Tom Duchamp, both of whom generously shared
their own notes and ideas about teaching this subject; Jim Isenberg and
Steve Mitchell, who had the courage to teach from these notes while they
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Preface
vii
were still in development, and who have provided spectacularly helpful
suggestions for improvement; and Gary Sandine, who after having found
an early version of these notes on the Web has read them with incredible
thoroughness and has made more suggestions than anyone else for improving them, and has even contributed several first-rate illustrations, with a
promise of more to come.
Happy reading!
John M. Lee
Seattle
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viii
Preface
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Contents
Preface
1 Smooth Manifolds
Topological Manifolds . . . . . . .
Smooth Structures . . . . . . . . .
Examples . . . . . . . . . . . . . .
Local Coordinate Representations
Manifolds With Boundary . . . . .
Problems . . . . . . . . . . . . . .
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2 Smooth Maps
Smooth Functions and Smooth Maps . .
Smooth Covering Maps . . . . . . . . .
Lie Groups . . . . . . . . . . . . . . . .
Bump Functions and Partitions of Unity
Problems . . . . . . . . . . . . . . . . .
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3 The Tangent Bundle
Tangent Vectors . . . . . . . . . . . . . . . . . . .
Push-Forwards . . . . . . . . . . . . . . . . . . . .
Computations in Coordinates . . . . . . . . . . . .
The Tangent Space to a Manifold With Boundary
Tangent Vectors to Curves . . . . . . . . . . . . . .
Alternative Definitions of the Tangent Space . . .
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Contents
The Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . .
Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 The Cotangent Bundle
Covectors . . . . . . . . . . . . .
Tangent Covectors on Manifolds
The Cotangent Bundle . . . . . .
The Differential of a Function . .
Pullbacks . . . . . . . . . . . . .
Line Integrals . . . . . . . . . . .
Conservative Covector Fields . .
Problems . . . . . . . . . . . . .
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5 Submanifolds
Submersions, Immersions, and Embeddings . . . .
Embedded Submanifolds . . . . . . . . . . . . . . .
The Inverse Function Theorem and Its Friends . .
Level Sets . . . . . . . . . . . . . . . . . . . . . . .
Images of Embeddings and Immersions . . . . . . .
Restricting Maps to Submanifolds . . . . . . . . .
Vector Fields and Covector Fields on Submanifolds
Lie Subgroups . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . .
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6 Embedding and Approximation Theorems
Sets of Measure Zero in Manifolds . . . . . . . .
The Whitney Embedding Theorem . . . . . . . .
The Whitney Approximation Theorem . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . .
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7 Lie Group Actions
Group Actions on Manifolds . . . . . . . .
Equivariant Maps . . . . . . . . . . . . . .
Quotients of Manifolds by Group Actions
Covering Manifolds . . . . . . . . . . . . .
Quotients of Lie Groups . . . . . . . . . .
Homogeneous Spaces . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . .
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8 Tensors
The Algebra of Tensors .
Tensors and Tensor Fields
Symmetric Tensors . . . .
Riemannian Metrics . . .
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. . . . . . . .
on Manifolds
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Contents
xi
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
9 Differential Forms
The Heuristics of Volume Measurement
The Algebra of Alternating Tensors . .
The Wedge Product . . . . . . . . . . .
Differential Forms on Manifolds . . . . .
Exterior Derivatives . . . . . . . . . . .
Symplectic Forms . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . .
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201
202
204
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214
219
225
10 Integration on Manifolds
Orientations . . . . . . . . . . . . . . .
Orientations of Hypersurfaces . . . . .
Integration of Differential Forms . . .
Stokes’s Theorem . . . . . . . . . . . .
Manifolds with Corners . . . . . . . .
Integration on Riemannian Manifolds
Problems . . . . . . . . . . . . . . . .
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229
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265
11 De Rham Cohomology
The de Rham Cohomology Groups .
Homotopy Invariance . . . . . . . . .
Computations . . . . . . . . . . . . .
The Mayer–Vietoris Theorem . . . .
Singular Homology and Cohomology
The de Rham Theorem . . . . . . .
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271
272
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285
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303
12 Integral Curves and Flows
Integral Curves . . . . . . . . . . . .
Flows . . . . . . . . . . . . . . . . .
The Fundamental Theorem on Flows
Complete Vector Fields . . . . . . .
Proof of the ODE Theorem . . . . .
Problems . . . . . . . . . . . . . . .
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307
307
309
314
316
317
325
13 Lie Derivatives
The Lie Derivative . . . . . . .
Lie Brackets . . . . . . . . . . .
Commuting Vector Fields . . .
Lie Derivatives of Tensor Fields
Applications . . . . . . . . . . .
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327
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www.pdfgrip.com
xii
Contents
14 Integral Manifolds and Foliations
Tangent Distributions . . . . . . . . .
Integral Manifolds and Involutivity . .
The Frobenius Theorem . . . . . . . .
Applications . . . . . . . . . . . . . . .
Foliations . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . .
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355
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369
15 Lie Algebras and Lie Groups
Lie Algebras . . . . . . . . . . . . . . .
Induced Lie Algebra Homomorphisms
One-Parameter Subgroups . . . . . . .
The Exponential Map . . . . . . . . .
The Closed Subgroup Theorem . . . .
Lie Subalgebras and Lie Subgroups . .
The Fundamental Correspondence . .
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371
371
378
381
385
392
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398
400
Appendix: Review of Prerequisites
403
Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
References
441
Index
445
www.pdfgrip.com
1
Smooth Manifolds
This book is about smooth manifolds. In the simplest terms, these are
spaces that locally look like some Euclidean space Rn , and on which one
can do calculus. The most familiar examples, aside from Euclidean spaces
themselves, are smooth plane curves such as circles and parabolas, and
smooth surfaces R3 such as spheres, tori, paraboloids, ellipsoids, and hyperboloids. Higher-dimensional examples include the set of unit vectors in
Rn+1 (the n-sphere) and graphs of smooth maps between Euclidean spaces.
You are probably already familiar with manifolds as examples of topological spaces: A topological manifold is a topological space with certain
properties that encode what we mean when we say that it “locally looks
like” Rn . Such spaces are studied intensively by topologists.
However, many (perhaps most) important applications of manifolds involve calculus. For example, the application of manifold theory to geometry
involves the study of such properties as volume and curvature. Typically,
volumes are computed by integration, and curvatures are computed by formulas involving second derivatives, so to extend these ideas to manifolds
would require some means of making sense of differentiation and integration
on a manifold. The application of manifold theory to classical mechanics involves solving systems of ordinary differential equations on manifolds, and
the application to general relativity (the theory of gravitation) involves
solving a system of partial differential equations.
The first requirement for transferring the ideas of calculus to manifolds is
some notion of “smoothness.” For the simple examples of manifolds we described above, all subsets of Euclidean spaces, it is fairly easy to describe
the meaning of smoothness on an intuitive level. For example, we might
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2
1. Smooth Manifolds
want to call a curve “smooth” if it has a tangent line that varies continuously from point to point, and similarly a “smooth surface” should be one
that has a tangent plane that varies continuously from point to point. But
for more sophisticated applications, it is an undue restriction to require
smooth manifolds to be subsets of some ambient Euclidean space. The ambient coordinates and the vector space structure of Rn are superfluous data
that often have nothing to do with the problem at hand. It is a tremendous advantage to be able to work with manifolds as abstract topological
spaces, without the excess baggage of such an ambient space. For example, in the application of manifold theory to general relativity, spacetime
is thought of as a 4-dimensional smooth manifold that carries a certain
geometric structure, called a Lorentz metric, whose curvature results in
gravitational phenomena. In such a model, there is no physical meaning
that can be assigned to any higher-dimensional ambient space in which the
manifold lives, and including such a space in the model would complicate
it needlessly. For such reasons, we need to think of smooth manifolds as
abstract topological spaces, not necessarily as subsets of larger spaces.
As we will see shortly, there is no way to define a purely topological
property that would serve as a criterion for “smoothness,” so topological
manifolds will not suffice for our purposes. As a consequence, we will think
of a smooth manifold as a set with two layers of structure: first a topology,
then a smooth structure.
In the first section of this chapter, we describe the first of these structures.
A topological manifold is a topological space with three special properties
that express the notion of being locally like Euclidean space. These properties are shared by Euclidean spaces and by all of the familiar geometric
objects that look locally like Euclidean spaces, such as curves and surfaces.
In the second section, we introduce an additional structure, called a
smooth structure, that can be added to a topological manifold to enable us
to make sense of derivatives. At the end of that section, we indicate how
the two-stage construction can be combined into a single step.
Following the basic definitions, we introduce a number of examples of
manifolds, so you can have something concrete in mind as you read the
general theory. (Most of the really interesting examples of manifolds will
have to wait until Chapter 5, however.) We then discuss in some detail how
local coordinates can be used to identify parts of smooth manifolds locally
with parts of Euclidean spaces. At the end of the chapter, we introduce
an important generalization of smooth manifolds, called manifolds with
boundary.
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Topological Manifolds
3
Topological Manifolds
This section is devoted to a brief overview of the definition and properties
of topological manifolds. We assume the reader is familiar with the basic
properties of topological spaces, at the level of [Lee00] or [Mun75], for
example.
Suppose M is a topological space. We say M is a topological manifold of
dimension n or a topological n-manifold if it has the following properties:
• M is a Hausdorff space: For every pair of points p, q ∈ M , there are
disjoint open subsets U, V ⊂ M such that p ∈ U and q ∈ V .
• M is second countable: There exists a countable basis for the topology
of M .
• M is locally Euclidean of dimension n: Every point has a neighborhood that is homeomorphic to an open subset of Rn .
The locally Euclidean property means that for each p ∈ M , we can find
the following:
• an open set U ⊂ M containing p;
• an open set U ⊂ Rn ; and
• a homeomorphism ϕ : U → U (i.e, a continuous bijective map with
continuous inverse).
Exercise 1.1. Show that equivalent definitions of locally Euclidean spaces
are obtained if, instead of requiring U to be homeomorphic to an open subset
of Rn , we require it to be homeomorphic to an open ball in Rn , or to Rn
itself.
The basic example of a topological n-manifold is, of course, Rn . It is
Hausdorff because it is a metric space, and it is second countable because
the set of all open balls with rational centers and rational radii is a countable basis.
Requiring that manifolds share these properties helps to ensure that
manifolds behave in the ways we expect from our experience with Euclidean
spaces. For example, it is easy to verify that in a Hausdorff space, onepoint sets are closed and limits of convergent sequences are unique. The
motivation for second countability is a bit less evident, but it will have
important consequences throughout the book, beginning with the existence
of partitions of unity in Chapter 2.
In practice, both the Hausdorff and second countability properties are
usually easy to check, especially for spaces that are built out of other manifolds, because both properties are inherited by subspaces and products, as
the following exercises show.
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4
1. Smooth Manifolds
Exercise 1.2. Show that any topological subspace of a Hausdorff space is
Hausdorff, and any finite product of Hausdorff spaces is Hausdorff.
Exercise 1.3. Show that any topological subspace of a second countable
space is second countable, and any finite product of second countable spaces
is second countable.
In particular, it follows easily from these two exercises that any open
subset of a topological n-manifold is itself a topological n-manifold (with
the subspace topology, of course).
One of the most important properties of second countable spaces is expressed the following lemma, whose proof can be found in [Lee00, Lemma
2.15].
Lemma 1.1. Let M be a second countable topological space. Then every
open cover of M has a countable subcover.
The way we have defined topological manifolds, the empty set is a topological n-manifold for every n. For the most part, we will ignore this special
case (sometimes without remembering to say so). But because it is useful
in certain contexts to allow the empty manifold, we have chosen not to
exclude it from the definition.
We should note that some authors choose to omit the the Hausdorff
property or second countability or both from the definition of manifolds.
However, most of the interesting results about manifolds do in fact require
these properties, and it is exceedingly rare to encounter a space “in nature”
that would be a manifold except for the failure of one or the other of these
hypotheses. See Problems 1-1 and 1-2 for a couple of examples.
Coordinate Charts
Let M be a topological n-manifold. A coordinate chart (or just a chart) on
M is a pair (U, ϕ), where U is an open subset of M and ϕ : U → U is a
homeomorphism from U to an open subset U = ϕ(U ) ⊂ Rn (Figure 1.1).
If in addition U is an open ball in Rn , then U is called a coordinate ball.
The definition of a topological manifold implies that each point p ∈ M is
contained in the domain of some chart (U, ϕ). If ϕ(p) = 0, we say the chart
is centered at p. Given p and any chart (U, ϕ) whose domain contains p,
it is easy to obtain a new chart centered at p by subtracting the constant
vector ϕ(p).
Given a chart (U, ϕ), we call the set U a coordinate domain, or a coordinate neighborhood of each of its points. The map ϕ is called a (local )
coordinate map, and the component functions of ϕ are called local coordinates on U . We will sometimes write things like “(U, ϕ) is a chart containing
p” as a shorthand for “(U, ϕ) is a chart whose domain U contains p.”
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Topological Manifolds
5
U
ϕ
U
FIGURE 1.1. A coordinate chart.
We conclude this section with a brief look at some examples of topological
manifolds.
Example 1.2 (Spheres). Let Sn denote the (unit ) n-sphere, which is
the set of unit-length vectors in Rn+1 :
Sn = {x ∈ Rn+1 : |x| = 1}.
It is Hausdorff and second countable because it is a subspace of Rn . To
show that it is locally Euclidean, for each index i = 1, . . . , n + 1, let Ui+
denote the subset of Sn where the ith coordinate is positive:
Ui+ = {(x1 , . . . , xn+1 ) ∈ Sn : xi > 0}.
Similarly, Ui− is the set where xi < 0.
±
n
For each such i, define maps ϕ±
i : Ui → R by
1
n+1
) = (x1 , . . . , xi , . . . , xn+1 ),
ϕ±
i (x , . . . , x
where the hat over xi indicates that xi is omitted. Each ϕ±
i is evidently a
continuous map, being the restriction to Sn of a linear map on Rn+1 . It is
a homeomorphism onto its image, the unit ball Bn ⊂ Rn , because it has a
continuous inverse given by
−1 1
(u , . . . , un ) = u1 , . . . , ui−1 , ±
(ϕ±
i )
1 − |u|2 , ui , . . . , un .
