An introduction to manifolds / Loring W. Tu.

Format:

Book

Author/Creator:

Tu, Loring W.

Series:

Universitext

Language:

English

Subjects (All):

Manifolds (Mathematics).

Physical Description:

xv, 360 pages : illustrations ; 24 cm.

Place of Publication:

New York : Springer, [2008]

Summary:

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, Introduction to Manifolds is also an excellent foundation for Springer GTM 82, Differential Forms in Algebraic Topology.

Contents:

Part I Euclidean Spaces

1 Smooth Functions on a Euclidean Space 5

1.1 C[infinity] Versus Analytic Functions 5

1.2 Taylor's Theorem with Remainder 7

2 Tangent Vectors in R[superscript n] as Derivations 11

2.1 The Directional Derivative 12

2.2 Germs of Functions 13

2.3 Derivations at a Point 14

2.4 Vector Fields 15

2.5 Vector Fields as Derivations 17

3 Alternating k-Linear Functions 19

3.1 Dual Space 19

3.2 Permutations 20

3.3 Multilinear Functions 22

3.4 Permutation Action on k-Linear Functions 23

3.5 The Symmetrizing and Alternating Operators 24

3.6 The Tensor Product 25

3.7 The Wedge Product 25

3.8 Anticommutativity of the Wedge Product 27

3.9 Associativity of the Wedge Product 28

3.10 A Basis for k-Covectors 30

4 Differential Forms on R[superscript n] 33

4.1 Differential 1-Forms and the Differential of a Function 33

4.2 Differential k-Forms 35

4.3 Differential Forms as Multilinear Functions on Vector Fields 36

4.4 The Exterior Derivative 36

4.5 Closed Forms and Exact Forms 39

4.6 Applications to Vector Calculus 39

4.7 Convention on Subscripts and Superscripts 42

Part II Manifolds

5 Manifolds 47

5.1 Topological Manifolds 47

5.2 Compatible Charts 48

5.3 Smooth Manifolds 50

5.4 Examples of Smooth Manifolds 51

6 Smooth Maps on a Manifold 57

6.1 Smooth Functions and Maps 57

6.2 Partial Derivatives 60

6.3 The Inverse Function Theorem 60

7 Quotients 63

7.1 The Quotient Topology 63

7.2 Continuity of a Map on a Quotient 64

7.3 Identification of a Subset to a Point 65

7.4 A Necessary Condition for a Hausdorff Quotient 65

7.5 Open Equivalence Relations 66

7.6 The Real Projective Space 68

7.7 The Standard C[infinity] Atlas on a Real Projective Space 71

Part III The Tangent Space

8 The Tangent Space 77

8.1 The Tangent Space at a Point 77

8.2 The Differential of a Map 78

8.3 The Chain Rule 79

8.4 Bases for the Tangent Space at a Point 80

8.5 Local Expression for the Differential 82

8.6 Curves in a Manifold 83

8.7 Computing the Differential Using Curves 85

8.8 Rank, Critical and Regular Points 86

9 Submanifolds 91

9.1 Submanifolds 91

9.2 The Zero Set of a Function 94

9.3 The Regular Level Set Theorem 95

9.4 Examples of Regular Submanifolds 97

10 Categories and Functors 101

10.1 Categories 101

10.2 Functors 102

10.3 Dual Maps 103

11 The Rank of a Smooth Map 105

11.1 Constant Rank Theorem 106

11.2 Immersions and Submersions 107

11.3 Images of Smooth Maps 109

11.4 Smooth Maps into a Submanifold 113

11.5 The Tangent Plane to a Surface in R[superscript 3] 115

12 The Tangent Bundle 119

12.1 The Topology of the Tangent Bundle 119

12.2 The Manifold Structure on the Tangent Bundle 121

12.3 Vector Bundles 121

12.4 Smooth Sections 123

12.5 Smooth Frames 125

13 Bump Functions and Partitions of Unity 127

13.1 C[infinity] Bump Functions 127

13.2 Partitions of Unity 131

13.3 Existence of a Partition of Unity 132

34 Vector Fields 135

14.1 Smoothness of a Vector Field 135

14.2 Integral Curves 136

14.3 Local Flows 138

14.4 The Lie Bracket 141

14.5 Related Vector Fields 143

14.6 The Push-Forward of a Vector Field 144

Part IV Lie Groups and Lie Algebras

15 Lie Groups 149

15.1 Examples of Lie Groups 149

15.2 Lie Subgroups 152

15.3 The Matrix Exponential 153

15.4 The Trace of a Matrix 155

15.5 The Differential of det at the Identity 157

16 Lie Algebras 161

16.1 Tangent Space at the Identity of a Lie Group 161

16.2 The Tangent Space to SL(n, R) at I 161

16.3 The Tangent Space to O(n) at I 162

16.4 Left-Invariant Vector Fields on a Lie Group 163

16.5 The Lie Algebra of a Lie Group 165

16.6 The Lie Bracket on gl(n, R) 166

16.7 The Push-Forward of a Left-Invariant Vector Field 167

16.8 The Differential as a Lie Algebra Homomorphism 168

Part V Differential Forms

17 Differential 1-Forms 175

17.1 The Differential of a Function 175

17.2 Local Expression for a Differential 1-Form 176

17.3 The Cotangent Bundle 177

17.4 Characterization of C[infinity] 1-Forms 177

17.5 Pullback of 1-forms 179

18 Differential k-Forms 181

18.1 Local Expression for a k-Form 182

18.2 The Bundle Point of View 183

18.3 C[infinity] k-Forms 183

18.4 Pullback of k-Forms 184

18.5 The Wedge Product 184

18.6 Invariant Forms on a Lie Group 186

19 The Exterior Derivative 189

19.1 Exterior Derivative on a Coordinate Chart 190

19.2 Local Operators 190

19.3 Extension of a Local Form to a Global Form 191

19.4 Existence of an Exterior Differentiation 192

19.5 Uniqueness of Exterior Differentiation 192

19.6 The Restriction of a k-Form to a Submanifold 193

19.7 A Nowhere-Vanishing 1-Form on the Circle 193

19.8 Exterior Differentiation Under a Pullback 195

Part VI Integration

20 Orientations 201

20.1 Orientations on a Vector Space 201

20.2 Orientations and n-Covectors 203

20.3 Orientations on a Manifold 204

20.4 Orientations and Atlases 206

21 Manifolds with Boundary 211

21.1 Invariance of Domain 211

21.2 Manifolds with Boundary 213

21.3 The Boundary of a Manifold with Boundary 215

21.4 Tangent Vectors, Differential Forms, and Orientations 215

21.5 Boundary Orientation for Manifolds of Dimension Greater than One 216

21.6 Boundary Orientation for One-Dimensional Manifolds 218

22 Integration on a Manifold 221

22.1 The Riemann Integral of a Function on R[superscript n] 221

22.2 Integrability Conditions 223

22.3 The Integral of an n-Form on R[superscript n] 224

22.4 The Integral of a Differential Form on a Manifold 225

22.5 Stokes' Theorem 228

22.6 Line Integrals and Green's Theorem 230

Part VII De Rham Theory

23 De Rham Cohomology 235

23.1 De Rham Cohomology 235

23.2 Examples of de Rham Cohomology 237

23.3 Diffeomorphism Invariance 239

23.4 The Ring Structure on de Rham Cohomology 240

24 The Long Exact Sequence in Cohomology 243

24.1 Exact Sequences 243

24.2 Cohomology of Cochain Complexes 245

24.3 The Connecting Homomorphism 246

24.4 The Long Exact Sequence in Cohomology 247

25 The Mayer-Vietoris Sequence 249

25.1 The Mayer-Vietoris Sequence 249

25.2 The Cohomology of the Circle 253

25.3 The Euler Characteristic 254

26 Homotopy Invariance 257

26.1 Smooth Homotopy 257

26.2 Homotopy Type 258

26.3 Deformation Retractions 260

26.4 The Homotopy Axiom for de Rham Cohomology 261

27 Computation of de Rham Cohomology 263

27.1 Cohomology Vector Space of a Torus 263

27.2 The Cohomology Ring of a Torus 265

27.3 The Cohomology of a Surface of Genus g 267

28 Proof of Homotopy Invariance 273

28.1 Reduction to Two Sections 274

28.2 Cochain Homotopies 274

28.3 Differential Forms on M x R 275

28.4 A Cochain Homotopy Between [Characters not reproducible] and [Characters not reproducible] 276

28.5 Verification of Cochain Homotopy 276

A Point-Set Topology 281

A.1 Topological Spaces 281

A.2 Subspace Topology 283

A.3 Bases 284

A.4 Second Countability 285

A.5 Separation Axioms 286

A.6 The Product Topology 287

A.7 Continuity 289

A.8 Compactness 290

A.9 Connectedness 293

A.10 Connected Components 294

A.11 Closure 295

A.12 Convergence 296

B The Inverse Function Theorem on R[superscript n] and Related Results 299

B.1 The Inverse Function Theorem 299

B.2 The Implicit Function Theorem 300

B.3 Constant Rank Theorem 303

C Existence of a Partition of Unity in General 307

D Linear Algebra 311

D.1 Linear Transformations 311

D.2 Quotient Vector Spaces 312

Solutions to Selected Exercises Within the Text 315

Hints and Solutions to Selected End-of-Chapter Problems 319.

Notes:

Includes bibliographical references (page [347]) and index.

ISBN:

9780387480985

0387480986

9780387481012

038748101X

OCLC:

186358733