Preface 1 Introduction 1.1 A Robot's Arm 1.2 The Configuration Space of Two Electrons 1.3 State Spaces and Fiber Bundles 1.4 Further Examples 1.5 Compact Surfaces 1.6 Higher Dimensions 2 Smooth Manifolds 2.1 Topological Manifolds 2.2 Smooth Structures 2.3 Maximal Atlases 2.4 Smooth Maps 2.5 Submanifolds 2.6 Products and Sums 3 The Tangent Space 3.1 Germs 3.2 Smooth Bump Functions 3.3 The Tangent Space 3.4 The Cotangent Space 3.5 Derivations 4 Regular Values 4.1 The Rank 4.2 The Inverse Function Theorem 4.3 The Rank Theorem 4.4 Regular Values 4.5 Transversality 4.6 Sard's Theorem 4.7 Immersions and Imbeddings 5 Vector Bundles 5.1 Topological Vector Bundles 5.2 Transition Functions 5.3 Smooth Vector Bundles 5.4 Pre-vector Bundles 5.5 The Tangent Bundle 5.6 The Cotangent Bundle 6 Constructions on Vector Bundles 6.1 Subbundles and Restrictions 6.2 The Induced Bundle 6.3 Whitney Sum of Bundles 6.4 Linear Algebra on Bundles 6.5 Normal Bundles 6.6 Riemannian Metrics 6.7 Orientations 6.8 The Generalized Gauss Map 7 Integrability 7.1 Flows and Velocity Fields 7.2 Integrability: Compact Case 7.3 Local Flows and Integrability 7.4 Second-Order Differential Equations
8 Local Phenomena that Go Global 8.1 Refinements of Covers 8.2 Partition of Unity 8.3 Global Properties of Smooth Vector Bundles 8.4 An Introduction to Morse Theory 8.5 Ehresmann's Fibration Theorem Appendix A Point Set Topology A.1 Topologies: Open and Closed Sets A.2 Continuous Maps A.3 Bases for Topologies A.4 Separation A.5 Subspaces A.6 Quotient Spaces A.7 Compact Spaces A.8 Product Spaces A.9 Connected Spaces A.10 Set-Theoretical Stuff Appendix B Hints or Solutions to the Exercises References Index
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