Spivak, Differential Geometry Vol. 5, 3rd Edition — Table of Contents

The complete table of contents of A Comprehensive Introduction to Differential Geometry, Volume V, 3rd Edition, by Michael Spivak (Publish or Perish, Inc.; ISBN 9780914098744), covering partial differential equations as geometers need them, existence and non-existence of isometric imbeddings, rigidity, and the generalized Gauss–Bonnet theorem with characteristic classes. Chapter and section headings are listed as they appear in the book.

Contents of Volume V

Chapter 1. And Now a Brief Message from Our Sponsor

• First order PDE's
  • Linear first order PDE's; characteristic curves; Cauchy problem for free initial curves
  • Quasi-linear first order PDE's; characteristic curves; Cauchy problem for free initial conditions; characteristic initial conditions
  • General first order PDE's; Monge cone; characteristic curves of a solution; characteristic strips; Cauchy problem for free initial data; characteristic initial data
  • First order PDE's in n variables
• Free initial manifolds for higher order equations
• Systems of first order PDE's
• The Cauchy–Kowalewski Theorem
• Classification of second order PDE's
  • Classification of semi-linear equations
  • Reduction to normal forms
  • Classification of general second order equations
• The prototypical PDE's of physics
  • The wave equation; the heat equation; Laplace's equation
  • Elementary properties
• Hyperbolic systems in two variables
• Hyperbolic second order equations in two variables
  • First reduction of the problem
  • New system of characteristic equations
  • Characteristic initial data
  • Monge–Ampère equations
• Elliptic solutions of second order equations in two variables
• Addendum: Differential systems; the Cartan–Kähler Theorem
• Addendum: An elementary maximum principle
• Problems

Chapter 2. Existence and Non-Existence of Isometric Imbeddings

• Non-imbeddability theorems; exteriorly orthogonal bilinear forms; index of nullity and index of relative nullity
• The Darboux equation
• Burstin–Janet–Cartan Theorem
• Addendum: The embedding problem via differential systems
• Problems

Chapter 3. Rigidity

• Rigidity in higher dimensions; type number
• Bendings, warpings, and infinitesimal bendings
• ℝ³-valued differential forms, the support function, and Minkowski's formulas
• Infinitesimal rigidity of convex surfaces
• Cohn-Vossen's Theorem
• Minkowski's Theorem
• Christoffel's Theorem
• Other problems, solved and unsolved
• Local problems; the role of the asymptotic curves
• Other classical results
• E. E. Levi's Theorems and Schilt's Theorem
• Surfaces in S³ and H³
• Rigidity for higher codimension
• Addendum: Infinitesimal bendings of rotation surfaces
• Problems

Chapter 4. The Generalized Gauss–Bonnet Theorem and What It Means for Mankind

• Historical remarks
• Operations on bundles
  • Bundle maps and principal bundle maps; Whitney sums and induced bundles; the covering homotopy theorem
• Grassmannians and universal bundles
• The Pfaffian
• Defining the Euler class in terms of a connection
  • The Euler class
  • The class C(ξ)
  • The Gauss–Bonnet–Chern Theorem
• The concept of characteristic classes
• The cohomology of homogeneous spaces
  • The C∞ structure of homogeneous spaces
  • Invariant forms
• A smattering of classical invariant theory
  • The Capelli identities
  • The first fundamental theorem of invariant theory for O(n) and SO(n)
• An easier invariance problem
• The cohomology of the oriented Grassmannians
  • Computation of the cohomology; Pontryagin classes
  • Describing the characteristic classes in terms of a connection
• The Weil homomorphism
• Complex bundles
  • Hermitian inner products, the unitary group, and complex Grassmannians
  • The cohomology of the complex Grassmannians; Chern classes
  • Relations between the Chern classes and the Pontryagin and Euler classes
• Valedictory
• Addendum: Invariant theory for the unitary group
• Addendum: Recovering the differential forms; the Gauss–Bonnet–Chern Theorem for manifolds-with-boundary

Bibliography

• Other topics in differential geometry
• Books
• Journal articles

Notation Index · Index

The Other Volumes

Tables of contents: Vol. 1 · Vol. 2 · Vol. 3 · Vol. 4. See also the Differential Geometry collection and the biography and complete bibliography of Michael Spivak.