Introduction to Global Analysis (eBook)

Front Cover 1
Introduction to Global Analysis 4
Copyright Page 5
Contents 6
Preface 10
Introduction 12
Chapter 1. Manifolds and Their Maps 16
Differentiable Manifolds and Their Maps 16
The Case of Euclidean Spaces 25
Power Series 34
Functions with Prescribed Properties Norms
Germs and Jets 44
Problems and Projects 47
Chapter 2. Embeddings and Immersions of Manifolds 49
Some Important Examples 50
The Tangent Space 58
Existence of Embeddings and Immersions 64
Approximation of Smooth Mappings 71
Problems and Projects 76
Chapter 3. Critical Values, Sard’s Theorem, and Transversality 79
Critical Points and Values 80
Sard’s Theorem Applications
Thom’s Transversality Lemma 90
Problems and Projects 96
Chapter 4. Tangent Bundles, Vector Bundles, and Classification 98
Groups Acting on Spaces 99
The Tangent Bundle 102
Vector Bundles 103
Constructions with Vector Bundles 105
The Classification of Vector Bundles 116
Examples of Classifications 140
Problems and Projects 145
Chapter 5. Differentiation and Integration on Manifolds 148
Integration in Several Variables 149
Exterior Algebra and Forms 152
Integration on Manifolds 167
The Poincarè Lemma 171
Stokes’ Theorem 174
Problems and Projects 180
Chapter 6. Differential Operators on Manifolds 182
Differential Operators on Smooth Bundles 183
Riemann Metrics and the Laplacian 188
Characterization of Linear Differential Operators 200
The Symbol Ellipticity
Problems and Projects 215
Chapter 7. Infinite-Dimensional Manifolds 217
Topological Vector Spaces 218
Elements of Infinite-Dimensional Manifolds 222
Hilbert Manifolds Partition of Unity
Function Spaces 228
The Unitary Group 231
Problems and Projects 235
Chapter 8. Morse Theory and Its Applications 237
Nondegenerate Critical Points 238
Homology and Morse Inequalities 244
Cell Decompositions from a Morse Function 258
Applications to Geodesics 260
Problems and Projects 265
Chapter 9. Lie Groups 267
Basic Theory of Lie Groups 268
The Idea of Lie Algebras 273
The Exponential Map 279
Closed Subgroups of Lie Groups 284
Invariant Forms and Integration 287
Representations of Lie Groups 291
Lie Groups Acting on Manifolds 296
Problems and Projects 298
Chapter 10. Dynamical Systems 300
Transformation Groups Invariant and Minimal Sets
Linear Differential Equations in Euclidean Space 308
Planar Flows: The Poincaè-Bendixson Theorem 313
Families of Subspaces The Frobenius Theorem
Problems and Projects 323
Chapter 11. A Description of Singularities and Catastrophes 325
Singularities of Smooth Maps Stability
Finite Determination and Codimension 330
Unfoldings of Singularities 333
Elementary Catastrophes 335
Bibliography 338
Index 344