A Brief Introduction 1
Chapter 1 Euclidean Spaces 3
1 Smooth Functions on a Euclidean Space 3
1.1 C∞ Versus Analytic Functions 4
1.2 Taylor's Theorem with Remainder 5
Problems 8
2 Tangent Vectors in Rn as Derivations 10
2.1 The Directional Derivative 10
2.2 Germs of Functions 11
2.3 Derivations at a Point 13
2.4 Vector Fields 14
2.5 Vector Fields as Derivations 16
Problems 17
3 The Exterior Algebra of Multicovectors 18
3.1 Dual Space 19
3.2 Permutations 20
3.3 Multilinear Functions 22
3.4 The Permutation Action on Multilinear Functions 23
3.5 The Symmetrizing and Alternating Operators 24
3.6 The Tensor Product 25
3.7 The Wedge Product 26
3.8 Anticommutativity of the Wedge Product 27
3.9 Associativity of the Wedge Product 28
3.10 A Basis for k-Covectors 31
Problems 32
4 Differential Forms on Rn 34
4.1 Differential 1-Forms and the Differential of a Function 34
4.2 Differential k-Forms 36
4.3 Differential Forms as Multilinear Functions on Vector Fields 37
4.4 The Exterior Derivative 38
4.5 Closed Forms and Exact Forms 40
4.6 Applications to Vector Calculus 41
4.7 Convention on Subscripts and Superscripts 44
Problems 44
Chapter 2 Manifolds 48
5 Manifolds 48
5.1 Topological Manifolds 48
5.2 Compatible Charts 49
5.3 Smooth Manifolds 52
5.4 Examples of Smooth Manifolds 53
Problems 57
6 Smooth Maps on a Manifold 59
6.1 Smooth Functions on a Manifold 59
6.2 Smooth Maps Between Manifolds 61
6.3 Diffeomorphisms 63
6.4 Smoothness in Terms of Components 63
6.5 Examples of Smooth Maps 65
6.6 Partial Derivatives 67
6.7 The Inverse Function Theorem 68
Problems 70
7 Quotients 71
7.1 The Quotient Topology 71
7.2 Continuity of a Map on a Quotient 72
7.3 Identification of a Subset to a Point 73
7.4 A Necessary Condition for a Hausdorff Quotient 73
7.5 Open Equivalence Relations 74
7.6 Real Projective Space 76
7.7 The Standard C∞ Atlas on a Real Projective Space 79
Problems 81
Chapter 3 The Tangent Space 86
8 The Tangent Space 86
8.1 The Tangent Space at a Point 86
8.2 The Differential of a Map 87
8.3 The Chain Rule 88
8.4 Bases for the Tangent Space at a Point 89
8.5 A Local Expression for the Differential 91
8.6 Curves in a Manifold 92
8.7 Computing the Differential Using Curves 95
8.8 Immersions and Submersions 96
8.9 Rank,and Critical and Regular Points 96
Problems 98
9 Submanifolds 100
9.1 Submanifolds 100
9.2 Level Sets of a Function 103
9.3 The Regular Level Set Theorem 105
9.4 Examples of Regular Submanifolds 106
Problems 108
10 Categories and Functors 110
10.1 Categories 110
10.2 Functors 111
10.3 The Dual Functor and the Multicovector Functor 113
Problems 114
11 The Rank of a Smooth Map 115
11.1 Constant Rank Theorem 115
11.2 The Immersion and Submersion Theorems 118
11.3 Images of Smooth Maps 120
11.4 Smooth Maps into a Submanifold 124
11.5 The Tangent Plane to a Surface in R3 125
Problems 127
12 The Tangent Bundle 129
12.1 The Topology of the Tangent Bundle 129
12.2 The Manifold Structure on the Tangent Bundle 132
12.3 Vector Bundles 133
12.4 Smooth Sections 136
12.5 Smooth Frames 137
Problems 139
13 Bump Functions and Partitions of Unity 140
13.1 C∞ Bump Functions 140
13.2 Partitions of Unity 145
13.3 Existence of a Partition of Unity 146
Problems 147
14 Vector Fields 149
14.1 Smoothness of a Vector Field 149
14.2 Integral Curves 152
14.3 Local Flows 154
14.4 The Lie Bracket 157
14.5 The Pushforward of Vector Fields 159
14.6 Related Vector Fields 159
Problems 161
Chapter 4 Lie Groups and Lie Algebras 164
15 Lie Groups 164
15.1 Examples of Lie Groups 164
15.2 Lie Subgroups 167
15.3 The Matrix Exponential 169
15.4 The Trace of a Matrix 171
15.5 The Differential of det at the Identity 174
Problems 174
16 Lie Algebras 178
16.1 Tangent Space at the Identity of a Lie Group 178
16.2 Left-Invariant Vector Fields on a Lie Group 180
16.3 The Lie Algebra of a Lie Group 182
16.4 The Lie Bracket on gl(n,R) 183
16.5 The Pushforward of Left-Invariant Vector Fields 184
16.6 The Differential as a Lie Algebra Homomorphism 185
Problems 187
Chapter 5 Differential Forms 190
17 Differential 1-Forms 190
17.1 The Differential of a Function 191
17.2 Local Expression for a Differential 1-Form 191
17.3 The Cotangent Bundle 192
17.4 Characterization of C∞ 1-Forms 193
17.5 Pullback of 1-Forms 195
17.6 Restriction of 1-Forms to an Immersed Submanifold 197
Problems 199
18 Differential k-Forms 200
18.1 Differential Forms 200
18.2 Local Expression for a k-Form 202
18.3 The Bundle Point of View 203
18.4 Smooth k-Forms 203
18.5 Pullback of k-Forms 204
18.6 The Wedge Product 205
18.7 Differential Forms on a Circle 206
18.8 Invariant Forms on a Lie Group 207
Problems 208
19 The Exterior Derivative 210
19.1 Exterior Derivative on a Coordinate Chart 211
19.2 Local Operators 211
19.3 Existence of an Exterior Derivative on a Manifold 212
19.4 Uniqueness of the Exterior Derivative 213
19.5 Exterior Differentiation Under a Pullback 214
19.6 Restriction of k-Forms to a Submanifold 216
19.7 A Nowhere-Vanishing 1-Form on the Circle 216
Problems 218
20 The Lie Derivative and Interior Multiplication 221
20.1 Families of Vector Fields and Differential Forms 221
20.2 The Lie Derivative of a Vector Field 223
20.3 The Lie Derivative of a Differential Form 226
20.4 Interior Multiplication 227
20.5 Properties of the Lie Derivative 229
20.6 Global Formulas for the Lie and Exterior Derivatives 232
Problems 233
Chapter 6 Integration 236
21 Orientations 236
21.1 Orientations of a Vector Space 236
21.2 Orientations and n-Covectors 238
21.3 Orientations on a Manifold 240
21.4 Orientations and Differential Forms 242
21.5 Orientations and Atlases 245
Problems 246
22 Manifolds with Boundary 248
22.1 Smooth Invariance of Domain in Rn 248
22.2 Manifolds with Boundary 250
22.3 The Boundary of a Manifold with Boundary 253
22.4 Tangent Vectors,Differential Forms,and Orientations 253
22.5 Outward-Pointing Vector Fields 254
22.6 Boundary Orientation 255
Problems 256
23 Integration on Manifolds 260
23.1 The Riemann Integral of a Function on Rn 260
23.2 Integrability Conditions 262
23.3 The Integral of an n-Form on Rn 263
23.4 Integral of a Differential Form over a Manifold 265
23.5 Stokes's Theorem 269
23.6 Line Integrals and Green's Theorem 271
Problems 272
Chapter 7 De Rham Theory 274
24 De Rham Cohomology 274
24.1 De Rham Cohomology 274
24.2 Examples of de Rham Cohomology 276
24.3 Diffeomorphism Invariance 278
24.4 The Ring Structure on de Rham Cohomology 279
Problems 280
25 The Long Exact Sequence in Cohomology 281
25.1 Exact Sequences 281
25.2 Cohomology of Cochain Complexes 283
25.3 The Connecting Homomorphism 284
25.4 The Zig-Zag Lemma 285
Problems 287
26 The Mayer-Vietoris Sequence 288
26.1 The Mayer-Vietoris Sequence 288
26.2 The Cohomology of the Circle 292
26.3 The Euler Characteristic 295
Problems 295
27 Homotopy Invariance 296
27.1 Smooth Homotopy 296
27.2 Homotopy Type 297
27.3 Deformation Retractions 299
27.4 The Homotopy Axiom for de Rham Cohomology 300
Problems 301
28 Computation of de Rham Cohomology 302
28.1 Cohomology Vector Space of a Torus 302
28.2 The Cohomology Ring of a Torus 303
28.3 The Cohomology of a Surface of Genus g 306
Problems 310
29 Proof of Homotopy Invariance 311
29.1 Reduction to Two Sections 311
29.2 Cochain Homotopies 312
29.3 Differential Forms on M×R 312
29.4 A Cochain Homotopy Between i？ and i？ 314
29.5 Verification of Cochain Homotopy 315
Problems 316
Appendices 317
A Point-Set Topology 317
A.1 Topological Spaces 317
A.2 Subspace Topology 320
A.3 Bases 321
A.4 First and Second Countability 323
A.5 Separation Axioms 324
A.6 Product Topology 326
A.7 Continuity 327
A.8 Compactness 329
A.9 Boundedness in Rn 332
A.10 Connectedness 332
A.11 Connected Components 333
A.12 Closure 334
A.13 Convergence 336
Problems 337
B The Inverse Function Theorem on Rn and Related Results 339
B.1 The Inverse Function Theorem 339
B.2 The Implicit Function Theorem 339
B.3 Constant Rank Theorem 343
Problems 344
C Existence of a Partition of Unity in General 346
D Linear Algebra 349
D.1 Quotient Vector Spaces 349
D.2 Linear Transformations 350
D.3 Direct Product and Direct Sum 351
Problems 352
E Quaternions and the Symplectic Group 353
E.1 Representation of Linear Maps by Matrices 354
E.2 Quaternionic Conjugation 355
E.3 Quaternionic Inner Product 356
E.4 Representations of Quaternions by Complex Numbers 356
E.5 Quaternionic Inner Product in Terms of Complex Components 357
E.6 H-Linearity in Terms of Complex Numbers 357
E.7 Symplectic Group 358
Problems 359
Solutions to Selected Exercises Within the Text 361
Hints and Solutions to Selected End-of-Section Problems 367
List of Notations 387
References 395
Index 397