流形导论 第2版 英文版PDF|Epub|txt|kindle电子书版本网盘下载

流形导论 第2版 英文版PDF格式电子书版下载

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A Brief Introduction1

Chapter 1 Euclidean Spaces3

1 Smooth Functions on a Euclidean Space3

1.1 C∞ Versus Analytic Functions4

1.2 Taylor's Theorem with Remainder5

Problems8

2 Tangent Vectors in Rn as Derivations10

2.1 The Directional Derivative10

2.2 Germs of Functions11

2.3 Derivations at a Point13

2.4 Vector Fields14

2.5 Vector Fields as Derivations16

Problems17

3 The Exterior Algebra of Multicovectors18

3.1 Dual Space19

3.2 Permutations20

3.3 Multilinear Functions22

3.4 The Permutation Action on Multilinear Functions23

3.5 The Symmetrizing and Alternating Operators24

3.6 The Tensor Product25

3.7 The Wedge Product26

3.8 Anticommutativity of the Wedge Product27

3.9 Associativity of the Wedge Product28

3.10 A Basis for k-Covectors31

Problems32

4 Differential Forms on Rn34

4.1 Differential 1-Forms and the Differential of a Function34

4.2 Differential k-Forms36

4.3 Differential Forms as Multilinear Functions on Vector Fields37

4.4 The Exterior Derivative38

4.5 Closed Forms and Exact Forms40

4.6 Applications to Vector Calculus41

4.7 Convention on Subscripts and Superscripts44

Problems44

Chapter 2 Manifolds48

5 Manifolds48

5.1 Topological Manifolds48

5.2 Compatible Charts49

5.3 Smooth Manifolds52

5.4 Examples of Smooth Manifolds53

Problems57

6 Smooth Maps on a Manifold59

6.1 Smooth Functions on a Manifold59

6.2 Smooth Maps Between Manifolds61

6.3 Diffeomorphisms63

6.4 Smoothness in Terms of Components63

6.5 Examples of Smooth Maps65

6.6 Partial Derivatives67

6.7 The Inverse Function Theorem68

Problems70

7 Quotients71

7.1 The Quotient Topology71

7.2 Continuity of a Map on a Quotient72

7.3 Identification of a Subset to a Point73

7.4 A Necessary Condition for a Hausdorff Quotient73

7.5 Open Equivalence Relations74

7.6 Real Projective Space76

7.7 The Standard C∞ Atlas on a Real Projective Space79

Problems81

Chapter 3 The Tangent Space86

8 The Tangent Space86

8.1 The Tangent Space at a Point86

8.2 The Differential of a Map87

8.3 The Chain Rule88

8.4 Bases for the Tangent Space at a Point89

8.5 A Local Expression for the Differential91

8.6 Curves in a Manifold92

8.7 Computing the Differential Using Curves95

8.8 Immersions and Submersions96

8.9 Rank,and Critical and Regular Points96

Problems98

9 Submanifolds100

9.1 Submanifolds100

9.2 Level Sets of a Function103

9.3 The Regular Level Set Theorem105

9.4 Examples of Regular Submanifolds106

Problems108

10 Categories and Functors110

10.1 Categories110

10.2 Functors111

10.3 The Dual Functor and the Multicovector Functor113

Problems114

11 The Rank of a Smooth Map115

11.1 Constant Rank Theorem115

11.2 The Immersion and Submersion Theorems118

11.3 Images of Smooth Maps120

11.4 Smooth Maps into a Submanifold124

11.5 The Tangent Plane to a Surface in R3125

Problems127

12 The Tangent Bundle129

12.1 The Topology of the Tangent Bundle129

12.2 The Manifold Structure on the Tangent Bundle132

12.3 Vector Bundles133

12.4 Smooth Sections136

12.5 Smooth Frames137

Problems139

13 Bump Functions and Partitions of Unity140

13.1 C∞ Bump Functions140

13.2 Partitions of Unity145

13.3 Existence of a Partition of Unity146

Problems147

14 Vector Fields149

14.1 Smoothness of a Vector Field149

14.2 Integral Curves152

14.3 Local Flows154

14.4 The Lie Bracket157

14.5 The Pushforward of Vector Fields159

14.6 Related Vector Fields159

Problems161

Chapter 4 Lie Groups and Lie Algebras164

15 Lie Groups164

15.1 Examples of Lie Groups164

15.2 Lie Subgroups167

15.3 The Matrix Exponential169

15.4 The Trace of a Matrix171

15.5 The Differential of det at the Identity174

Problems174

16 Lie Algebras178

16.1 Tangent Space at the Identity of a Lie Group178

16.2 Left-Invariant Vector Fields on a Lie Group180

16.3 The Lie Algebra of a Lie Group182

16.4 The Lie Bracket on gl(n,R)183

16.5 The Pushforward of Left-Invariant Vector Fields184

16.6 The Differential as a Lie Algebra Homomorphism185

Problems187

Chapter 5 Differential Forms190

17 Differential 1-Forms190

17.1 The Differential of a Function191

17.2 Local Expression for a Differential 1-Form191

17.3 The Cotangent Bundle192

17.4 Characterization of C∞ 1-Forms193

17.5 Pullback of 1-Forms195

17.6 Restriction of 1-Forms to an Immersed Submanifold197

Problems199

18 Differential k-Forms200

18.1 Differential Forms200

18.2 Local Expression for a k-Form202

18.3 The Bundle Point of View203

18.4 Smooth k-Forms203

18.5 Pullback of k-Forms204

18.6 The Wedge Product205

18.7 Differential Forms on a Circle206

18.8 Invariant Forms on a Lie Group207

Problems208

19 The Exterior Derivative210

19.1 Exterior Derivative on a Coordinate Chart211

19.2 Local Operators211

19.3 Existence of an Exterior Derivative on a Manifold212

19.4 Uniqueness of the Exterior Derivative213

19.5 Exterior Differentiation Under a Pullback214

19.6 Restriction of k-Forms to a Submanifold216

19.7 A Nowhere-Vanishing 1-Form on the Circle216

Problems218

20 The Lie Derivative and Interior Multiplication221

20.1 Families of Vector Fields and Differential Forms221

20.2 The Lie Derivative of a Vector Field223

20.3 The Lie Derivative of a Differential Form226

20.4 Interior Multiplication227

20.5 Properties of the Lie Derivative229

20.6 Global Formulas for the Lie and Exterior Derivatives232

Problems233

Chapter 6 Integration236

21 Orientations236

21.1 Orientations of a Vector Space236

21.2 Orientations and n-Covectors238

21.3 Orientations on a Manifold240

21.4 Orientations and Differential Forms242

21.5 Orientations and Atlases245

Problems246

22 Manifolds with Boundary248

22.1 Smooth Invariance of Domain in Rn248

22.2 Manifolds with Boundary250

22.3 The Boundary of a Manifold with Boundary253

22.4 Tangent Vectors,Differential Forms,and Orientations253

22.5 Outward-Pointing Vector Fields254

22.6 Boundary Orientation255

Problems256

23 Integration on Manifolds260

23.1 The Riemann Integral of a Function on Rn260

23.2 Integrability Conditions262

23.3 The Integral of an n-Form on Rn263

23.4 Integral of a Differential Form over a Manifold265

23.5 Stokes's Theorem269

23.6 Line Integrals and Green's Theorem271

Problems272

Chapter 7 De Rham Theory274

24 De Rham Cohomology274

24.1 De Rham Cohomology274

24.2 Examples of de Rham Cohomology276

24.3 Diffeomorphism Invariance278

24.4 The Ring Structure on de Rham Cohomology279

Problems280

25 The Long Exact Sequence in Cohomology281

25.1 Exact Sequences281

25.2 Cohomology of Cochain Complexes283

25.3 The Connecting Homomorphism284

25.4 The Zig-Zag Lemma285

Problems287

26 The Mayer-Vietoris Sequence288

26.1 The Mayer-Vietoris Sequence288

26.2 The Cohomology of the Circle292

26.3 The Euler Characteristic295

Problems295

27 Homotopy Invariance296

27.1 Smooth Homotopy296

27.2 Homotopy Type297

27.3 Deformation Retractions299

27.4 The Homotopy Axiom for de Rham Cohomology300

Problems301

28 Computation of de Rham Cohomology302

28.1 Cohomology Vector Space of a Torus302

28.2 The Cohomology Ring of a Torus303

28.3 The Cohomology of a Surface of Genus g306

Problems310

29 Proof of Homotopy Invariance311

29.1 Reduction to Two Sections311

29.2 Cochain Homotopies312

29.3 Differential Forms on M×R312

29.4 A Cochain Homotopy Between i？ and i？314

29.5 Verification of Cochain Homotopy315

Problems316

Appendices317

A Point-Set Topology317

A.1 Topological Spaces317

A.2 Subspace Topology320

A.3 Bases321

A.4 First and Second Countability323

A.5 Separation Axioms324

A.6 Product Topology326

A.7 Continuity327

A.8 Compactness329

A.9 Boundedness in Rn332

A.10 Connectedness332

A.11 Connected Components333

A.12 Closure334

A.13 Convergence336

Problems337

B The Inverse Function Theorem on Rn and Related Results339

B.1 The Inverse Function Theorem339

B.2 The Implicit Function Theorem339

B.3 Constant Rank Theorem343

Problems344

C Existence of a Partition of Unity in General346

D Linear Algebra349

D.1 Quotient Vector Spaces349

D.2 Linear Transformations350

D.3 Direct Product and Direct Sum351

Problems352

E Quaternions and the Symplectic Group353

E.1 Representation of Linear Maps by Matrices354

E.2 Quaternionic Conjugation355

E.3 Quaternionic Inner Product356

E.4 Representations of Quaternions by Complex Numbers356

E.5 Quaternionic Inner Product in Terms of Complex Components357

E.6 H-Linearity in Terms of Complex Numbers357

E.7 Symplectic Group358

Problems359

Solutions to Selected Exercises Within the Text361

Hints and Solutions to Selected End-of-Section Problems367

List of Notations387

References395

Index397