Lee introduction to smooth manifolds( GTM 218)

You are probably already familiar with manifolds as examples of logical spaces: A topological manifold is a topological space with certainproperties that encode what we mean when we say

INTRODUCTION TO SMOOTH MANIFOLDS

by John M Lee

University of Washington Department of Mathematics

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.”

• Page 13, Example 1.11: Just before and in the displayed equation, change ϕ± j ◦ (ϕ± i )−1 to

ϕ±

i ◦ (ϕ± j)−1 (twice).

• Page 21, Problem 1-3: Change the definition of eσ to eσ(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

defin-ition of smooth maps between manifolds given here, because this defindefin-ition 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)

con-taining p and (V, ψ) concon-taining 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 f 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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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:

evaf = d

dt

t=0

f (a + tv).

• Page 51, line 15: Insert “p ↔ bp,” 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} × Rk∼ = Rk.”

• 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

• Page 64, Problem 3-2: The third displayed equation should be

α−1(X

1, , Xk) = j1∗X1+ · · · + jk∗Xk.

• Page 74, second line from bottom: Delete the symbol γ∗.

• Page 83, last displayed equation: Should be changed to

• Page 86, Example 4.26, line 1: Change “Example 4.7” to “Example 4.18.”

• Page 95, part (f): Change “(y − 2)2+ z2+ 1” to “(y − 2)2+ z2 = 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

“topol-ogy.”

• 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



vj ∂

∂xj

x



• Page 142, three lines above the last displayed equation: Change “a eδ-approximation”

to “e δ-close.”

3

I → N.”

• 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 6= ∅}”

to “contained in GK 0 = {g ∈ G : (g · K0) ∩ K0 6= ∅}, where K0= 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 Tστ to Tτ σ.

• Page 183, first line after the third display: Change “η = στ” to “η = τσ.”

• 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

sym-plectic 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.”

4

• Page 239, second display: Two occurrences of dxi should be changed to dx1, so the

• 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

• Page 256, equation (10.10): Change σ1 and σ0 to eσ1 and eσ0, respectively.

• Pages 259–267: Change every occurrence of h·, ·i to h·, ·ig.

• Page 261, proof of Lemma 10.38, 7th line: Change “Corollary 10.40” to “Proposition

• 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

• 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.”

5

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: f M → M” to “α: f M → f M ”

• Page 284, line 5: Change y < R to y < −R, and change e 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

“sin-gular 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))

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 Tijdxi to

Y Tijdxi⊗ dxj.

• Page 343, proof of Lemma 13.17, last line: Change (LVW ) to ( LXτ ).

• Page 343, proof of Proposition 13.18, first line: Change “θ∗

tτ = τ for all t” to “θ∗

tτ = 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 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)n i.

• 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.”

8

This book is an introductory graduate-level textbook on the theory ofsmooth manifolds, for students who already have a solid acquaintance withgeneral topology, the fundamental group, and covering spaces, as well asbasic 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 avoidedthis term, however, because it applies more properly to the study of smoothmanifolds endowed with some extra structure, such as a Riemannian met-ric, a symplectic structure, a Lie group structure, or a foliation, and of theproperties that are invariant under maps that preserve the structure Al-though I do treat all of these subjects in this book, they are treated more asinteresting examples to which to apply the general theory than as objects

of study in their own right A student who finishes this book should bewell prepared to go on to study any of these specialized subjects in muchgreater depth

The book is organized roughly as follows Chapters 1 through 4 aremainly definitions It is the bane of this subject that there are so manydefinitions that must be piled on top of one another before anything in-teresting can be said, much less proved I have tried, nonetheless, to bring

in significant applications as early and as often as possible The first onecomes at the end of Chapter 4, where I show how to generalize the classicaltheory of line integrals to manifolds

The next three chapters, 5 through 7, present the first of four majorfoundational theorems on which all of smooth manifolds theory rests—theinverse function theorem—and some applications of it: to submanifold the-

ory, embeddings of smooth manifolds into Euclidean spaces, approximation

of continuous maps by smooth ones, and quotients of manifolds by groupactions

The next four chapters, 8 through 11, focus on tensors and tensor fields

on manifolds, and progress from Riemannian metrics through differentialforms, integration, and Stokes’s theorem (the second of the four founda-tional theorems), culminating in the de Rham theorem, which relates dif-ferential forms on a smooth manifold to its topology via its singular coho-mology 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 ofideas surrounding integral curves and flows of vector fields, which are thesmooth-manifold version of systems of ordinary differential equations Iprove a basic version of the existence, uniqueness, and smoothness theo-rem for ordinary differential equations in Chapter 12, and use that to provethe 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 tional theorems), and to explore the relationship between Lie groups andLie algebras

founda-The Appendix (which most readers should read first, or at least skim)contains a very cursory summary of prerequisite material on linear algebraand calculus that is used throughout the book One large piece of prereq-uisite material that should probably be in the Appendix, but is not yet,

is a summary of general topology, including the theory of the fundamentalgroup and covering spaces If you need a review of that, you will have tolook at another book (Of course, I recommend [Lee00], but there are manyother texts that will serve at least as well!)

This is still a work in progress, and there are bound to be errors andomissions Thus you will have to be particularly alert for typos and othermistakes Please let me know as soon as possible when you find any errors,unclear descriptions, or questionable statements I’ll post corrections onthe 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 togenerate them Any instructor teaching from this book should be sure todraw all the relevant pictures in class, and any student studying from themshould make an effort to draw pictures whenever possible

Acknowledgments There are many people who have contributed to the

de-velopment of this book in indispensable ways I would like to mention pecially Judith Arms and Tom Duchamp, both of whom generously sharedtheir own notes and ideas about teaching this subject; Jim Isenberg andSteve Mitchell, who had the courage to teach from these notes while they

Happy reading!

Topological Manifolds 3

Smooth Structures 6

Examples 11

Local Coordinate Representations 18

Manifolds With Boundary 19

Problems 21

2 Smooth Maps 23 Smooth Functions and Smooth Maps 24

Smooth Covering Maps 28

Lie Groups 30

Bump Functions and Partitions of Unity 34

Problems 40

3 The Tangent Bundle 41 Tangent Vectors 42

Push-Forwards 46

Computations in Coordinates 49

The Tangent Space to a Manifold With Boundary 52

Tangent Vectors to Curves 53

Alternative Definitions of the Tangent Space 55

The Tangent Bundle 57

Vector Fields 60

Problems 64

4 The Cotangent Bundle 65 Covectors 65

Tangent Covectors on Manifolds 68

The Cotangent Bundle 69

The Differential of a Function 71

Pullbacks 75

Line Integrals 78

Conservative Covector Fields 82

Problems 90

5 Submanifolds 93 Submersions, Immersions, and Embeddings 94

Embedded Submanifolds 97

The Inverse Function Theorem and Its Friends 105

Level Sets 113

Images of Embeddings and Immersions 118

Restricting Maps to Submanifolds 121

Vector Fields and Covector Fields on Submanifolds 122

Lie Subgroups 124

Problems 126

6 Embedding and Approximation Theorems 129 Sets of Measure Zero in Manifolds 130

The Whitney Embedding Theorem 133

The Whitney Approximation Theorem 138

Problems 144

7 Lie Group Actions 145 Group Actions on Manifolds 145

Equivariant Maps 149

Quotients of Manifolds by Group Actions 152

Covering Manifolds 157

Quotients of Lie Groups 160

Homogeneous Spaces 161

Problems 167

8 Tensors 171 The Algebra of Tensors 172

Tensors and Tensor Fields on Manifolds 179

Symmetric Tensors 182

Riemannian Metrics 184

Contents xi

Problems 196

9 Differential Forms 201 The Heuristics of Volume Measurement 202

The Algebra of Alternating Tensors 204

The Wedge Product 208

Differential Forms on Manifolds 212

Exterior Derivatives 214

Symplectic Forms 219

Problems 225

10 Integration on Manifolds 229 Orientations 230

Orientations of Hypersurfaces 235

Integration of Differential Forms 240

Stokes’s Theorem 248

Manifolds with Corners 251

Integration on Riemannian Manifolds 257

Problems 265

11 De Rham Cohomology 271 The de Rham Cohomology Groups 272

Homotopy Invariance 274

Computations 277

The Mayer–Vietoris Theorem 285

Singular Homology and Cohomology 291

The de Rham Theorem 297

Problems 303

12 Integral Curves and Flows 307 Integral Curves 307

Flows 309

The Fundamental Theorem on Flows 314

Complete Vector Fields 316

Proof of the ODE Theorem 317

Problems 325

13 Lie Derivatives 327 The Lie Derivative 327

Lie Brackets 329

Commuting Vector Fields 335

Lie Derivatives of Tensor Fields 339

Applications 343

Problems 352

14 Integral Manifolds and Foliations 355

Tangent Distributions 356

Integral Manifolds and Involutivity 357

The Frobenius Theorem 359

Applications 361

Foliations 364

Problems 369

15 Lie Algebras and Lie Groups 371 Lie Algebras 371

Induced Lie Algebra Homomorphisms 378

One-Parameter Subgroups 381

The Exponential Map 385

The Closed Subgroup Theorem 392

Lie Subalgebras and Lie Subgroups 394

The Fundamental Correspondence 398

Problems 400

Appendix: Review of Prerequisites 403 Linear Algebra 403

Calculus 424

Smooth Manifolds

This book is about smooth manifolds In the simplest terms, these arespaces that locally look like some Euclidean space Rn, and on which onecan do calculus The most familiar examples, aside from Euclidean spacesthemselves, are smooth plane curves such as circles and parabolas, andsmooth surfaces R3 such as spheres, tori, paraboloids, ellipsoids, and hy-perboloids 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 logical spaces: A topological manifold is a topological space with certainproperties that encode what we mean when we say that it “locally lookslike”Rn Such spaces are studied intensively by topologists

topo-However, many (perhaps most) important applications of manifolds volve calculus For example, the application of manifold theory to geometryinvolves the study of such properties as volume and curvature Typically,volumes are computed by integration, and curvatures are computed by for-mulas involving second derivatives, so to extend these ideas to manifoldswould require some means of making sense of differentiation and integration

on a manifold The application of manifold theory to classical mechanics volves solving systems of ordinary differential equations on manifolds, andthe application to general relativity (the theory of gravitation) involvessolving a system of partial differential equations

in-The first requirement for transferring the ideas of calculus to manifolds issome notion of “smoothness.” For the simple examples of manifolds we de-scribed above, all subsets of Euclidean spaces, it is fairly easy to describethe meaning of smoothness on an intuitive level For example, we might

want to call a curve “smooth” if it has a tangent line that varies ously from point to point, and similarly a “smooth surface” should be onethat has a tangent plane that varies continuously from point to point Butfor more sophisticated applications, it is an undue restriction to requiresmooth manifolds to be subsets of some ambient Euclidean space The am-bient coordinates and the vector space structure ofRnare superfluous datathat often have nothing to do with the problem at hand It is a tremen-dous advantage to be able to work with manifolds as abstract topologicalspaces, without the excess baggage of such an ambient space For exam-ple, in the application of manifold theory to general relativity, spacetime

continu-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 meaningthat can be assigned to any higher-dimensional ambient space in which themanifold lives, and including such a space in the model would complicate

it needlessly For such reasons, we need to think of smooth manifolds asabstract topological spaces, not necessarily as subsets of larger spaces

As we will see shortly, there is no way to define a purely topologicalproperty that would serve as a criterion for “smoothness,” so topologicalmanifolds 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 propertiesthat express the notion of being locally like Euclidean space These prop-erties are shared by Euclidean spaces and by all of the familiar geometricobjects that look locally like Euclidean spaces, such as curves and surfaces

In the second section, we introduce an additional structure, called asmooth 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 howthe two-stage construction can be combined into a single step

Following the basic definitions, we introduce a number of examples ofmanifolds, so you can have something concrete in mind as you read thegeneral theory (Most of the really interesting examples of manifolds willhave to wait until Chapter 5, however.) We then discuss in some detail howlocal coordinates can be used to identify parts of smooth manifolds locallywith parts of Euclidean spaces At the end of the chapter, we introduce

an important generalization of smooth manifolds, called manifolds withboundary

Topological Manifolds 3Topological 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 basicproperties of topological spaces, at the level of [Lee00] or [Mun75], forexample

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

neighbor-hood that is homeomorphic to an open subset ofRn

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 e U ⊂ R n; and

• a homeomorphism ϕ: U → e 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

ofRn, we require it to be homeomorphic to an open ball in Rn, or toRn

itself

The basic example of a topological n-manifold is, of course, Rn It isHausdorff because it is a metric space, and it is second countable becausethe set of all open balls with rational centers and rational radii is a count-able basis

Requiring that manifolds share these properties helps to ensure thatmanifolds behave in the ways we expect from our experience with Euclideanspaces For example, it is easy to verify that in a Hausdorff space, one-point sets are closed and limits of convergent sequences are unique Themotivation for second countability is a bit less evident, but it will haveimportant consequences throughout the book, beginning with the existence

of partitions of unity in Chapter 2

In practice, both the Hausdorff and second countability properties areusually easy to check, especially for spaces that are built out of other man-ifolds, because both properties are inherited by subspaces and products, asthe following exercises show

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 pressed the following lemma, whose proof can be found in [Lee00, Lemma2.15]

ex-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

topo-logical 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 toexclude it from the definition

We should note that some authors choose to omit the the Hausdorffproperty or second countability or both from the definition of manifolds.However, most of the interesting results about manifolds do in fact requirethese 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 thesehypotheses 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 → e U is a

homeomorphism from U to an open subset e U = ϕ(U ) ⊂ R n (Figure 1.1)

If in addition eU is an open ball inRn , 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

co-ordinate neighborhood of each of its points The map ϕ is called a (local ) coordinate map, and the component functions of ϕ are called local coordi- nates 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.”

Topological Manifolds 5

U

e

U ϕ

FIGURE 1.1 A coordinate chart

We conclude this section with a brief look at some examples of topologicalmanifolds

Example 1.2 (Spheres). Let Sn denote the (unit ) n-sphere, which is

the set of unit-length vectors inRn+1:

Sn

={x ∈ R n+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 U i+

denote the subset ofSn where the ith coordinate is positive:

U i+={(x1, , x n+1)∈ S n

: x i > 0}.

Similarly, U i − is the set where x i < 0.

For each such i, define maps ϕ ± i : U i ± → R n by

ϕ ± i (x1, , x n+1 ) = (x1, , b x i , , x n+1 ), where the hat over x i indicates that x i is omitted Each ϕ ± i is evidently acontinuous map, being the restriction toSn of a linear map onRn+1 It is

a homeomorphism onto its image, the unit ballBn ⊂ R n, because it has acontinuous inverse given by

Since every point inSn+1 is in the domain of one of these 2n + 2 charts,Sn

is locally Euclidean of dimension n and is thus a topological n-manifold.

Example 1.3 (Projective Spaces). The n-dimensional real projective

space, denoted by Pn (or sometimes RPn), is defined as the set of dimensional linear subspaces of Rn+1 We give it the quotient topology

1-determined by the natural map π : Rn+1 r {0} → P n

sending each point

x ∈ R n+1 r {0} to the line through x and 0 For any point x ∈ R n+1 r {0}, let [x] = π(x) denote the equivalence class of x inPn

For each i = 1, , n + 1, let e U i ⊂ R n+1 r {0} be the set where x i 6= 0,

and let U i = π( e U i)⊂ P n Since eU i is a saturated open set (meaning that it

contains the full inverse image π −1 (π(p)) for each p ∈ e U i ), U i is open and

π : e U i → U i is a quotient map Define a map ϕ i : U i → R n

This map is well-defined because its value is unchanged by multiplying x

by a nonzero constant, and it is continuous because ϕ i ◦ π is continuous.

(The characteristic property of a quotient map π is that a map f from the quotient space is continuous if and only if the composition f ◦ π is

continuous; see [Lee00].) In fact, ϕ iis a homeomorphism, because its inverse

is given by

ϕ −1 i (u1, , u n ) = [u1, , u i −1 , 1, u i , , u n ],

as you can easily check Geometrically, if we identifyRn

in the obvious way

with the affine subspace where x i = 1, then ϕ i [x] can be interpreted as the point where the line [x] intersects this subspace Because the sets U i cover

Pn, this shows thatPn is locally Euclidean of dimension n The Hausdorff

and second countability properties are left as exercises

Exercise 1.4 Show that Pn is Hausdorff and second countable, and is

therefore a topological n-manifold.

Smooth Structures

The definition of manifolds that we gave in the preceding section is cient for studying topological properties of manifolds, such as compactness,connectedness, simple connectedness, and the problem of classifying man-ifolds up to homeomorphism However, in the entire theory of topologicalmanifolds, there is no mention of calculus There is a good reason for this:Whatever sense we might try to make of derivatives of functions or curves

suffi-on a manifold, they cannot be invariant under homeomorphisms For

ex-ample, if f is a function on the circleS1, we would want to consider f to

Smooth Structures 7

be differentiable if it has an ordinary derivative with respect to the

an-gle θ But the circle is homeomorphic to the unit square, and because of

the corners the homeomorphism and its inverse cannot simultaneously bedifferentiable Thus, depending on the homeomorphism we choose, therewill either be functions on the circle whose composition with the homeo-morphism is not differentiable on the square, or vice versa (Although thisclaim may seem plausible, it is probably not obvious at this point how toprove it After we have developed some more machinery, you will be asked

to prove it in Problem 5-11.)

To make sense of derivatives of functions, curves, or maps, we will need

to introduce a new kind of manifold called a “smooth manifold.”

(Through-out this book, we will use the word “smooth” to mean C ∞, or infinitelydifferentiable.)

From the example above, it is clear that we cannot define a smoothmanifold simply to be a topological manifold with some special property,because the property of “smoothness” (whatever that might be) cannot beinvariant under homeomorphisms

Instead, we are going to define a smooth manifold as one with someextra structure in addition to its topology, which will allow us to decidewhich functions on the manifold are smooth To see what this additional

structure might look like, consider an arbitrary topological n-manifold M Each point in M is in the domain of a coordinate map ϕ : U → e U ⊂ R n

A plausible definition of a smooth function on M would be to say that

f : M → R is smooth if and only if the composite function f ◦ ϕ −1: eU → R

is smooth But this will make sense only if this property is independent

of the choice of coordinate chart To guarantee this, we will restrict ourattention to “smooth charts.” Since smoothness is not a homeomorphism-invariant property, the way to do this is to consider the collection of all

smooth charts as a new kind of structure on M In the remainder of this

chapter, we will carry out the details

Our study of smooth manifolds will be based on the calculus of maps

between Euclidean spaces If U and V are open subsets of Euclidean spaces

Rn and Rm , respectively, a map F : U → V is said to be smooth if each

of the component functions of F has continuous partial derivatives of all orders If in addition F is bijective and has a smooth inverse map, it is called

a diffeomorphism A diffeomorphism is, in particular, a homeomorphism A

review of some of the most important properties of smooth maps is given

in the Appendix

Let M be a topological n-manifold If (U, ϕ), (V, ψ) are two charts such that U ∩ V 6= ∅, then the composite map ψ ◦ ϕ −1 : ϕ(U ∩ V ) → ψ(U ∩ V )

(called the transition map from ϕ to ψ) is a composition of

homeomor-phisms, and is therefore itself a homeomorphism (Figure 1.2) Two charts

(U, ϕ) and (V, ψ) are said to be smoothly compatible if either U ∩ V = ∅

or the transition map ψ ◦ ϕ −1 is a diffeomorphism (Since ϕ(U ∩ V ) and ψ(U ∩ V ) are open subsets of R n

, smoothness of this map is to be

FIGURE 1.2 A transition map.

preted in the ordinary sense of having continuous partial derivatives of allorders.)

We define an atlas for M to be a collection of charts whose domains cover

M An atlas A is called a smooth atlas if any two charts in A are smoothly

compatible with each other

In practice, to show that the charts of an atlas are smoothly compatible,

it suffices to check that the transition map ψ ◦ ϕ −1 is smooth for everypair of coordinate maps ϕ and ψ, for then reversing the roles of ϕ and ψ shows that the inverse map (ψ ◦ ϕ −1)−1 = ϕ ◦ ψ −1 is also smooth, so each

transition map is in fact a diffeomorphism We will use this observationwithout further comment in what follows

Our plan is to define a “smooth structure” on M by giving a smooth atlas, and to define a function f : M → R to be smooth if and only if f ◦ ϕ −1 is

smooth (in the ordinary sense of functions defined on open subsets ofRn

)

for each coordinate chart (U, ϕ) in the atlas There is one minor technical

problem with this approach: In general, there will be many possible choices

of atlas that give the “same” smooth structure, in that they all determine

the same collection of smooth functions on M For example, consider the

Smooth Structures 9

following pair of atlases onRn:

A1={(R n , Id)}

A2={(B1(x), Id) : x ∈ R n },

where B1(x) is the unit ball around x and Id is the identity map Although

these are different smooth atlases, clearly they determine the same tion of smooth functions on the manifoldRn(namely, those functions thatare smooth in the sense of ordinary calculus)

collec-We could choose to define a smooth structure as an equivalence class ofsmooth atlases under an appropriate equivalence relation However, it ismore straightforward to make the following definition A smooth atlasA

on M is maximal if it is not contained in any strictly larger smooth atlas.

This just means every chart that is smoothly compatible with every chart

inA is already in A (Such a smooth atlas is also said to be complete.) Now we can define the main concept of this chapter A smooth structure

on a topological n-manifold M is a maximal smooth atlas A smooth

mani-fold is a pair (M, A), where M is a topological manifold and A is a smooth structure on M When the smooth structure is understood, we usually omit mention of it and just say “M is a smooth manifold.” Smooth structures are also called differentiable structures or C ∞ structures by some authors We

will use the term smooth manifold structure to mean a manifold topology

together with a smooth structure

We emphasize that a smooth structure is an additional piece of datathat must be added to a topological manifold before we are entitled to talkabout a “smooth manifold.” In fact, a given topological manifold may havemany different smooth structures (we will return to this issue in the nextchapter) And it should be noted that it is not always possible to find anysmooth structure—there exist topological manifolds that admit no smoothstructures at all

It is worth mentioning that the notion of smooth structure can be eralized in several different ways by changing the compatibility require-ment for charts For example, if we replace the requirement that charts besmoothly compatible by the weaker requirement that each transition map

gen-ψ ◦ ϕ −1 (and its inverse) be of class C k , we obtain the definition of a C k

structure Similarly, if we require that each transition map be real-analytic

(i.e., expressible as a convergent power series in a neighborhood of each

point), we obtain the definition of a real-analytic structure, also called a

C ω structure If M has even dimension n = 2m, we can identify R2mwith

Cm

and require that the transition maps be complex analytic; this

deter-mines a complex analytic structure A manifold endowed with one of these structures is called a C k manifold, real-analytic manifold, or complex man- ifold, respectively (Note that a C0manifold is just a topological manifold.)

We will not treat any of these other kinds of manifolds in this book, butthey play important roles in analysis, so it is useful to know the definitions

Without further qualification, every manifold mentioned in this bookwill be assumed to be a smooth manifold endowed with a specific smoothstructure In particular examples, the smooth structure will usually be

obvious from the context If M is a smooth manifold, any chart contained

in the given maximal smooth atlas will be called a smooth chart, and the corresponding coordinate map will be called a smooth coordinate map.

It is generally not very convenient to define a smooth structure by plicitly describing a maximal smooth atlas, because such an atlas contains

ex-very many charts Fortunately, we need only specify some smooth atlas, as

the next lemma shows

Lemma 1.4 Let M be a topological manifold.

(a) Every smooth atlas for M is contained in a unique maximal smooth

atlas.

(b) Two smooth atlases for M determine the same maximal smooth atlas

if and only if their union is a smooth atlas.

Proof Let A be a smooth atlas for M, and let A denote the set of all charts

that are smoothly compatible with every chart inA To show that A is asmooth atlas, we need to show that any two charts ofA are compatible with

each other, which is to say that for any (U, ϕ), (V, ψ) ∈ A, ψ ◦ ϕ −1 : ϕ(U ∩

V ) → ψ(U ∩ V ) is smooth.

Let x = ϕ(p) ∈ ϕ(U ∩V ) be arbitrary Because the domains of the charts

inA cover M, there is some chart (W, θ) ∈ A such that p ∈ W Since every

chart inA is smoothly compatible with (W, θ), both the maps θ ◦ ϕ −1 and

ψ ◦ θ −1 are smooth where they are defined Since p ∈ U ∩ V ∩ W , it follows

that ψ ◦ϕ −1 = (ψ ◦θ −1)◦(θ ◦ϕ −1 ) is smooth on a neighborhood of x Thus

ψ ◦ ϕ −1 is smooth in a neighborhood of each point in ϕ(U ∩ V ) Therefore

A is a smooth atlas To check that it is maximal, just note that any chartthat is smoothly compatible with every chart in A must in particular besmoothly compatible with every chart in A, so it is already in A Thisproves the existence of a maximal smooth atlas containingA If B is anyother maximal smooth atlas containingA, each of its charts is smoothlycompatible with each chart inA, so B ⊂ A By maximality of B, B = A.

The proof of (b) is left as an exercise

Exercise 1.5 Prove Lemma 1.4(b).

For example, if a topological manifold M can be covered by a single

chart, the smooth compatibility condition is trivially satisfied, so any such

chart automatically determines a smooth structure on M

Examples 11Examples

Before proceeding further with the general theory, let us establish someexamples of smooth manifolds

Example 1.5 (Euclidean spaces). Rn is a smooth n-manifold with the

smooth structure determined by the atlas consisting of the single chart(Rn , Id) We call this the standard smooth structure, and the resulting co-

ordinate map standard coordinates Unless we explicitly specify otherwise,

we will always use this smooth structure onRn

Example 1.6 (Finite-dimensional vector spaces). Let V be any

finite-dimensional vector space Any norm on V determines a topology,

which is independent of the choice of norm (Exercise A.21 in the

Appen-dix) With this topology, V has a natural smooth structure defined as lows Any (ordered) basis (E1, , E n ) for V defines a linear isomorphism

This map is a homeomorphism, so the atlas consisting of the single chart

(V, E −1) defines a smooth structure To see that this smooth structure

is independent of the choice of basis, let ( eE1, , e E n) be any other basisand let eE(x) =P

j x j Eejbe the corresponding isomorphism There is some

invertible matrix (A j i ) such that E i =P

j A j i Eej for each j The transitionmap between the two charts is then given by eE −1 ◦ E(x) = ex, where

i A j i x i Thus the map from x to ex is an invertible

linear map and hence a diffeomorphism, so the two charts are smoothlycompatible This shows that the union of the two charts determined byany two bases is still a smooth atlas, and thus all bases determine the same

smooth structure We will call this the standard smooth structure on V The Einstein Summation Convention

This is a good place to pause and introduce an important notational vention that we will use throughout the book Because of the proliferation

con-of summations such as P

i x i E i in this subject, we will often abbreviatesuch a sum by omitting the summation sign, as in

E(x) = x i E i

We interpret any such expression according to the following rule, called

the Einstein summation convention: If the same index name (such as i in

the expression above) appears twice in any term, once as an upper indexand once as a lower index, that term is understood to be summed overall possible values of that index, generally from 1 to the dimension of thespace in question This simple idea was introduced by Einstein to reducethe complexity of the expressions arising in the study of smooth manifolds

by eliminating the necessity of explicitly writing summation signs

Another important aspect of the summation convention is the positions

of the indices We will always write basis vectors (such as E i) with lower

indices, and components of a vector with respect to a basis (such as x i) withupper indices These index conventions help to ensure that, in summationsthat make mathematical sense, any index to be summed over will typicallyappear twice in any given term, once as a lower index and once as an upperindex

To be consistent with our convention of writing components of vectorswith upper indices, we need to use upper indices for the coordinates of

a point (x1, , x n) ∈ R n, and we will do so throughout this book though this may seem awkward at first, in combination with the summa-tion convention it offers enormous advantages when working with compli-cated indexed sums, not the least of which is that expressions that are notmathematically meaningful often identify themselves quickly by violatingthe index convention (The main exceptions are the Euclidean dot product

Al-x·y =P

i x i y i , in which i appears twice as an upper index, and certain

expressions involving matrices We will always explicitly write summationsigns in such expressions.)

More Examples

Now we continue with our examples of smooth manifolds

Example 1.7 (Matrices) Let M(m × n, R) denote the space of m × n

matrices with real entries It is a vector space of dimension mn under trix addition and scalar multiplication Thus M(m × n, R) is a smooth mn-

ma-dimensional manifold Similarly, the space M(m × n, C) of m × n complex

matrices is a vector space of dimension 2mn overR, and thus a manifold

of dimension 2mn In the special case m = n (square matrices), we will abbreviate M(n × n, R) and M(n × n, C) by M(n, R) and M(n, C), respec-

Examples 13

More generally, let M be a smooth n-manifold and U ⊂ M any open

subset Define an atlas on U by

AU ={smooth charts (V, ϕ) for M such that V ⊂ U}.

It is easy to verify that this is a smooth atlas for U Thus any open subset

of a smooth n-manifold is itself a smooth n-manifold in a natural way We call such a subset an open submanifold of M

Example 1.9 (The General Linear Group). The general linear group GL(n, R) is the set of invertible n × n matrices with real entries.

It is an n2-dimensional manifold because it is an open subset of the n2

-dimensional vector space M(n,R), namely the set where the (continuous)determinant function is nonzero

Example 1.10 (Matrices of Maximal Rank) The previous example

has a natural generalization to rectangular matrices of maximal rank

Sup-pose m < n, and let M m (m × n, R) denote the subset of M(m × n, R)

consisting of matrices of rank m If A is an arbitrary such matrix, the fact that rank A = m means that A has some nonsingular m × m minor By

continuity of the determinant function, this same minor has nonzero

de-terminant on some neighborhood of A in M(m × n, R), which implies that

A has a neighborhood contained in M m (m × n, R) Thus M m (m × n, R) is

an open subset of M (m × n, R), and therefore is itself an mn-dimensional

manifold A similar argument shows that Mn (m × n, R) is an mn-manifold

when n < m.

Exercise 1.6 If k is an integer between 0 and min(m, n), show that the

set of m × n matrices whose rank is at least k is an open submanifold of

M(m × n, R).

Example 1.11 (Spheres) We showed in Example 1.2 that the n-sphere

Sn ⊂ R n+1 is a topological n-manifold We put a smooth structure onSn

as follows For each i = 1, , n + 1, let (U i ± , ϕ ± i ) denote the coordinate

chart we constructed in Example 1.2 For any distinct indices i and j, the transition map ϕ ± j ◦ (ϕ ±

i )−1 is easily computed In the case i < j, we get

is a smooth atlas, and so defines a smooth structure on Sn We call this

its standard smooth structure The coordinates defined above will be called

graph coordinates, because they arise from considering the sphere locally

as the graph of the function u i=±p1− |u|2

Exercise 1.7 By identifying R2 with C in the usual way, we can think

of the unit circleS1as a subset of the complex plane An angle function on

a subset U ⊂ S1 is a continuous function θ : U → R such that e iθ (p) = p

for all p ∈ U Show that there exists an angle function θ on an open subset

U ⊂ S1 if and only if U 6= S1 For any such angle function, show that (U, θ)

is a smooth coordinate chart forS1 with its standard smooth structure.

Example 1.12 (Projective spaces) The n-dimensional real projective

space Pn is a topological n-manifold by Example 1.3 We will show that the coordinate charts (U i , ϕ i) constructed in that example are all smoothly

compatible Assuming for convenience that i > j, it is straightforward to

which is a diffeomorphism from ϕ i (U i ∩ U j ) to ϕ j (U i ∩ U j)

Example 1.13 (Product Manifolds). Suppose M1, , M k are

smooth manifolds of dimensions n1, , n k respectively The product

space M1× · · · × M k is Hausdorff by Exercise 1.2 and second countable

by Exercise 1.3 Given a smooth chart (U i , ϕ i ) for each M i, the map

ϕ1× · · · × ϕ k : U1× · · · × U k → R n1+···+n k is a homeomorphism onto itsimage, which is an open subset of Rn1+···+n k Thus the product set is a

topological manifold of dimension n1+· · · + n k, with charts of the form

(U1×· · ·×U k , ϕ1×· · ·×ϕ k) Any two such charts are smoothly compatiblebecause, as is easily verified,

(ψ1× · · · × ψ k)◦ (ϕ1× · · · × ϕ k)−1 = (ψ1◦ ϕ −1

1 )× · · · × (ψ k ◦ ϕ −1

k ),

which is a smooth map This defines a natural smooth manifold structure

on the product, called the product smooth manifold structure For example, this yields a smooth manifold structure on the n-dimensional torusTn =

S1× · · · × S1

In each of the examples we have seen so far, we have constructed a smoothmanifold structure in two stages: We started with a topological space andchecked that it was a topological manifold, and then we specified a smoothstructure It is often more convenient to combine these two steps into asingle construction, especially if we start with a set or a topological spacethat is not known a priori to be a topological manifold The followinglemma provides a shortcut

Lemma 1.14 (One-Step Smooth Manifold Structure) Let M be a

set, and suppose we are given a collection {U α } of subsets of M, together with an injective map ϕ α : U α → R n

for each α, such that the following properties are satisfied.

Examples 15

(i) For each α, e U α = ϕ α (U α ) is an open subset ofRn

(ii) For each α and β, ϕ α (U α ∩ U β ) and ϕ β (U α ∩ U β ) are open inRn

(iii ) Whenever U α ∩ U β 6= ∅, ϕ β ◦ ϕ −1

α : ϕ α (U α ∩ U β)→ ϕ β (U α ∩ U β ) is

smooth.

(iv ) Countably many of the sets U α cover M

(v ) Whenever p, q are distinct points in M , either there exists some U α

containing both p and q or there exist disjoint sets U α , U β with p ∈ U α

and q ∈ U β

Then M has a unique smooth manifold structure such that each (U α , ϕ α)

is a smooth chart.

Proof We define the topology by taking the sets of the form ϕ −1 α (V ), where

V ⊂ e U αis open, as a basis To prove that this is a basis for a topology, let

ϕ −1 α (V ) and ϕ −1 β (W ) be two such basis sets Properties (ii) and (iii) imply that ϕ α ◦ ϕ −1

β (W ) is an open subset of ϕ α (U α ∩ U β), and therefore also ofe

U α Thus if p is any point in ϕ −1 α (V ) ∩ ϕ −1 β (W ), then

ϕ −1 α (V ∩ ϕ α ◦ ϕ −1

β (W )) = ϕ −1 α (V ) ∩ ϕ −1

β (W )

is a basis open set containing p Each of the maps ϕ αis then a

homeomor-phism (essentially by definition), so M is locally Euclidean of dimension n.

If {U α i } is a countable collection of the sets U α covering M , each of the sets U α i has a countable basis, and the union of all these is a countable

basis for M , so M is second countable, and the Hausdorff property follows

easily from (v) Finally, (iii) guarantees that the collection{(U α , ϕ α)} is a

smooth atlas It is clear that this topology and smooth structure are theunique ones satisfying the conclusions of the lemma

Example 1.15 (Grassmann Manifolds). Let V be an n-dimensional

real vector space For any integer 0≤ k ≤ n, we let G k (V ) denote the set

of all k-dimensional linear subspaces of V We will show that G k (V ) can be naturally given the structure of a smooth manifold of dimension k(n − k).

The construction is somewhat more involved than the ones we have done

so far, but the basic idea is just to use linear algebra to construct charts for

Gk (V ) and then use Lemma 1.14 to show that these charts yield a smooth

manifold structure Since we will give a more straightforward proof that

Gk (V ) is a smooth manifold after we have developed more machinery in

Chapter 7, you may skip the details of this construction on first reading ifyou wish

Let P and Q be any complementary subspaces of V of dimensions k and (n −k), respectively, so that V decomposes as a direct sum: V = P ⊕Q The

graph of any linear map A : P → Q is a k-dimensional subspace Γ(A) ⊂ V ,

defined by

Γ(A) = {x + Ax : x ∈ P }.

Any such subspace has the property that its intersection with Q is the zero

subspace Conversely, any subspace with this property is easily seen to be

the graph of a unique linear map A : P → Q.

Let L(P, Q) denote the vector space of linear maps from P to Q, and let U Q denote the subset of Gk (V ) consisting of k-dimensional subspaces whose intersection with Q is trivial Define a map ψ : L(P, Q) → U Q by

ψ(A) = Γ(A).

The discussion above shows that ψ is a bijection Let ϕ = ψ −1 : U Q → L(P, Q) By choosing bases for P and Q, we can identify L(P, Q) with

M((n −k)×k, R) and hence with R k(n−k) , and thus we can think of (U Q , ϕ)

as a coordinate chart Since the image of each chart is all of L(P, Q),

con-dition (i) of Lemma 1.14 is clearly satisfied

Now let (P 0 , Q 0 ) be any other such pair of subspaces, and let ψ 0 , ϕ 0 be

the corresponding maps The set ϕ(U Q ∩ U Q 0) ⊂ L(P, Q) consists of all

A ∈ L(P, Q) whose graphs intersect both Q and Q 0trivially, which is easily

seen to be an open set, so (ii) holds We need to show that the transition

map ϕ 0 ◦ ϕ −1 = ϕ 0 ◦ ψ is smooth on this set This is the trickiest part of

the argument

Suppose A ∈ ϕ(U Q ∩ U Q 0)⊂ L(P, Q) is arbitrary, and let S denote the

subspace ψ(A) = Γ(A) ⊂ V If we put A 0 = ϕ 0 ◦ψ(A), then A 0is the uniquelinear map from P 0 to Q 0 whose graph is equal to S To identify this map, let x 0 ∈ P 0 be arbitrary, and note that A 0 x 0 is the unique element of Q 0 such that x 0 + A 0 x 0 ∈ S, which is to say that

x 0 + A 0 x 0 = x + Ax for some x ∈ P (1.1)

(See Figure 1.3.) There is in fact a unique x ∈ P for which this holds,

characterized by the property that

x + Ax − x 0 ∈ Q 0 .

If we let I A : P → V denote the map I A (x) = x + Ax and let π P 0 : V → P 0

be the projection onto P 0 with kernel Q 0 , then x satisfies

0 = π P 0 (x + Ax − x 0 ) = π

P 0 ◦ I A (x) − x 0 .

As long as A stays in the open subset of maps whose graphs intersect both

Q and Q 0 trivially, π P 0 ◦ I A : P → P 0 is invertible, and thus we can solvethis last equation for x to obtain x = (π P 0 ◦ I A)−1 (x 0 ) Therefore, A 0 is

given in terms of A by

A 0 x 0 = I x − x 0 = I ◦ (π 0 ◦ I )−1 (x 0)− x 0 . (1.2)

FIGURE 1.3 Smooth compatibility of coordinates on Gk (V ).

If we choose bases (E i 0 ) for P 0 and (F j 0 ) for Q 0, the columns of the matrix

representation of A 0 are the components of A 0 E i 0 By (1.2), this can bewritten

A 0 E i 0 = I A ◦ (π P 0 ◦ I A)−1 (E i 0)− E 0

i

The matrix entries of I A clearly depend smoothly on those of A, and thus so also do those of π P 0 ◦I A By Cramer’s rule, the components of the inverse of

a matrix are rational functions of the matrix entries, so the expression above

shows that the components of A 0 E i 0 depend smoothly on the components

of A This proves that ϕ 0 ◦ ϕ −1 is a smooth map, so the charts we have

constructed satisfy condition (iii) of Lemma 1.14

To check the countability condition (iv), we just note that Gk (V ) can in fact be covered by finitely many of the sets U Q : For example, if (E1, , E n)

is any fixed basis for V , any partition of the basis elements into two subsets containing k and n − k elements determines appropriate subspaces P and

Q, and any subspace S must have trivial intersection with Q for at least

one of these partitions (see Exercise A.4) Thus Gk (V ) is covered by the

finitely many charts determined by all possible partitions of a fixed basis.Finally, the Hausdorff condition (v) is easily verified by noting that for any

two k-dimensional subspaces P, P 0 ⊂ V , it is possible to find a subspace Q

of dimension n − k whose intersections with both P and P 0 are trivial, andthen P and P 0 are both contained in the domain of the chart determined

by, say, (P, Q).

The smooth manifold Gk (V ) is called the Grassmann manifold of planes in V , or simply a Grassmannian In the special case V =Rn, theGrassmannian Gk(Rn

k-) is often denoted by some simpler notation such as

Gk,n or G(k, n) Note that G1(Rn+1 ) is exactly the n-dimensional projective

spacePn

e

U ϕ

FIGURE 1.4 A coordinate grid

Exercise 1.8 Let 0 < k < n be integers, and let P, Q ⊂ R n

be the

subspaces spanned by (e1, , e k ) and (e k+1, , e n ), respectively, where e i

is the ith standard basis vector For any k-dimensional subspace S ⊂ R n that has trivial intersection with Q, show that the coordinate representation

ϕ(S) constructed in the preceding example is the unique (n −k)×k matrix B

such that S is spanned by the columns of the matrixÿI

k

B

þ

, where I kdenotes

the k × k identity matrix.

Local Coordinate Representations

Here is how one usually thinks about local coordinate charts on a smooth

manifold Once we choose a chart (U, ϕ) on M , the coordinate map ϕ : U →

e

U ⊂ R n can be thought of as giving an identification between U and e U

Using this identification, we can think of U simultaneously as an open subset of M and (at least temporarily while we work with this chart) as

an open subset ofRn You can visualize this identification by thinking of a

“grid” drawn on U representing the inverse images of the coordinate lines under ϕ (Figure 1.4) Under this identification, we can represent a point

p ∈ M by its coordinates (x1, , x n ) = ϕ(p), and think of this n-tuple as

being the point p We will typically express this by saying “(x1, , x n) is

the (local) coordinate representation for p” or “p = (x1, , x n) in localcoordinates.”

Manifolds With Boundary 19

FIGURE 1.5 A manifold with boundary

Another way to look at it is that by means of our identification U ↔ e U ,

we can think of ϕ as the identity map and suppress it from the notation.

This takes a bit of getting used to, but the payoff is a huge simplification

of the notation in many situations You just need to remember that theidentification depends heavily on the choice of coordinate chart

For example, if M = R2, let U = {(x, y) : x > 0} be the open right

half-plane, and let ϕ : U → R2 be the polar coordinate map ϕ(x, y) = (r, θ) = (p

x2+ y2, arctan y/x) We can write a given point p ∈ U

ei-ther as p = (x, y) in standard coordinates or as p = (r, θ) in

po-lar coordinates, where the two coordinate representations are related by

(r, θ) = (p

x2+ y2, arctan y/x) and (x, y) = (r cos θ, r sin θ).

Manifolds With Boundary

For some purposes, we will need the following generalization of manifolds

An n-dimensional topological manifold with boundary is a second

count-able Hausdorff space in which every point has a neighborhood

homeo-morphic to an open subset of the closed n-dimensional upper half space

Hn ={(x1, , x n)∈ R n : x n ≥ 0} (Figure 1.5) An open subset U ⊂ M

together with a homeomorphism ϕ from U to an open subset ofHnis called

a generalized chart for M

some generalized chart is called a boundary point of M , and a point that is

in the inverse image of IntHn

is called an interior point The boundary of

M (the set of all its boundary points) is denoted ∂M ; similarly its interior

is denoted Int M

Be careful to observe the distinction between this use of the terms

“boundary” and “interior” and their usage to refer to the boundary and

interior of a subset of a topological space A manifold M with boundary

may have nonempty boundary in this new sense, irrespective of whether ithas a boundary as a subset of some other topological space If we need toemphasize the difference between the two notions of boundary, we will use

the terms topological boundary or manifold boundary as appropriate.

To see how to define a smooth structure on a manifold with boundary,

recall that a smooth map from an arbitrary subset A ⊂ R n is defined

to be one that extends smoothly to an open neighborhood of A (see the Appendix) Thus if U is an open subset ofHn

, a smooth map F : U → R k

is a map that extends to a smooth map eF : e U → R k

, where eU is some open

a fact (which we will neither prove nor use) that F : U → R k has such a

smooth extension if and only if F is continuous, F | U ∩Int H n is smooth, and

each of the partial derivatives of F | U ∩Int H n has a continuous extension to

U ∩ H n

For example, let B2 ⊂ R2 denote the unit disk, let U = B2∩ H2, and

define f : U → R by f(x, y) =p1− x2− y2 Because f extends to all of

B2(by the same formula), f is a smooth function on U On the other hand, although g(x, y) = √

y is continuous on U and smooth in U ∩ Int H2, it has

no smooth extension to any neighborhood of U inR2because ∂g/∂y → ∞

as y → 0 Thus g is not a smooth function on U.

Given a topological manifold with boundary M , we define an atlas for

M as before to be a collection of generalized charts whose domains cover

M Two such charts (U, ϕ), (V, ψ) are smoothly compatible if ψ ◦ ϕ −1 is

smooth (in the sense just described) wherever it is defined Just as in the

case of manifolds, a smooth atlas for M is an atlas all of whose charts are smoothly compatible with each other, and a smooth structure for M is a

maximal smooth atlas

It can be shown using homology theory that the interior and boundary

of a topological manifold with boundary are disjoint (see [Lee00, Problem13-9], for example) We will not need this result, because the analogousresult for smooth manifolds with boundary is much easier to prove (or will

be, after we have developed a bit more machinery) A proof is outlined inProblem 5-19

Since any open ball in Rn

admits a diffeomorphism onto an open set ofHn

sub-, a smooth n-manifold is automatically a smooth n-manifold with

boundary (whose boundary is empty), but the converse is not true: A ifold with boundary is a manifold if and only if its boundary is empty (Thiswill follow from the fact that interior points and boundary points are dis-tinct.)

Instead, we are going to define a smooth manifold as one with someextra structure in addition to its topology, which will allow us to decidewhich functions on the manifold are smooth To see what this... added to a topological manifold before we are entitled to talkabout a ? ?smooth manifold.” In fact, a given topological manifold may havemany different smooth structures (we will return to this... chapter A smooth structure

on a topological n-manifold M is a maximal smooth atlas A smooth

mani-fold is a pair (M, A), where M is a topological manifold and A is a smooth