Spivak, Differential Geometry Vol. 1, 3rd Edition — Table of Contents

The complete table of contents of A Comprehensive Introduction to Differential Geometry, Volume I, 3rd Edition, by Michael Spivak (Publish or Perish, Inc.; ISBN 9780914098706), covering manifolds, differential structures, tensors, vector fields, differential forms, integration, Riemannian metrics, Lie groups, and an excursion into algebraic topology. Chapter and section headings are listed as they appear in the book.

This volume is in print, hardcover, direct from the publisher: A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition. All five volumes are also available as a complete set.

Contents of Volume I

Chapter 1. Manifolds

  • Elementary properties of manifolds
  • Examples of manifolds
  • Problems

Chapter 2. Differential Structures

  • C∞ structures
  • C∞ functions
  • Partial derivatives
  • Critical points
  • Immersion theorems
  • Partitions of unity
  • Problems

Chapter 3. The Tangent Bundle

  • The tangent space of ℝⁿ
  • The tangent space of an imbedded manifold
  • Vector bundles
  • The tangent bundle of a manifold
  • Equivalence classes of curves, and derivations
  • Vector fields
  • Orientation
  • Addendum: Equivalence of tangent bundles
  • Problems

Chapter 4. Tensors

  • The dual bundle
  • The differential of a function
  • Classical versus modern terminology
  • Multilinear functions
  • Covariant and contravariant tensors
  • Mixed tensors, and contraction
  • Problems

Chapter 5. Vector Fields and Differential Equations

  • Integral curves
  • Existence and uniqueness theorems
  • The local flow
  • One-parameter groups of diffeomorphisms
  • Lie derivatives
  • Brackets
  • Addendum: Differential equations
  • Addendum: Parameter curves in two dimensions
  • Problems

Chapter 6. Integral Manifolds

  • Prologue; classical integrability theorems
  • Local theory; Frobenius integrability theorem
  • Global theory
  • Problems

Chapter 7. Differential Forms

  • Alternating functions
  • The wedge product
  • Forms
  • Differential of a form
  • Frobenius integrability theorem (second version)
  • Closed and exact forms
  • The Poincaré Lemma
  • Problems

Chapter 8. Integration

  • Classical line and surface integrals
  • Integrals over singular k-cubes
  • The boundary of a chain
  • Stokes' Theorem
  • Integrals over manifolds
  • Volume elements
  • Stokes' Theorem
  • de Rham cohomology
  • Problems

Chapter 9. Riemannian Metrics

  • Inner products
  • Riemannian metrics
  • Length of curves
  • The calculus of variations
  • The First Variation Formula and geodesics
  • The exponential map
  • Geodesic completeness
  • Addendum: Tubular neighborhoods
  • Problems

Chapter 10. Lie Groups

  • Lie groups
  • Left invariant vector fields
  • Lie algebras
  • Subgroups and subalgebras
  • Homomorphisms
  • One-parameter subgroups
  • The exponential map
  • Closed subgroups
  • Left invariant forms
  • Bi-invariant metrics
  • The equations of structure
  • Problems

Chapter 11. Excursion in the Realm of Algebraic Topology

  • Complexes and exact sequences
  • The Mayer–Vietoris sequence
  • Triangulations
  • The Euler characteristic
  • Mayer–Vietoris sequence for compact supports
  • The exact sequence of a pair
  • Poincaré Duality
  • The Thom class
  • Index of a vector field
  • Poincaré–Hopf Theorem
  • Problems

Appendix A

  • Additional problems for Chapters 1, 2, 6, 7 and 8

Notation Index · Index

The Other Volumes

Tables of contents: Vol. 2 · Vol. 3 · Vol. 4 · Vol. 5. See also the Differential Geometry collection and the biography and complete bibliography of Michael Spivak.