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

The complete table of contents of A Comprehensive Introduction to Differential Geometry, Volume III, 3rd Edition, by Michael Spivak (Publish or Perish, Inc.; ISBN 9780914098720), covering the fundamental equations for hypersurfaces, the classical theory of surfaces, a compendium of surfaces, curves on surfaces, complete surfaces of constant curvature, and the Gauss–Bonnet theorem. Chapter and section headings are listed as they appear in the book.

Contents of Volume III

Chapter 1. The Fundamental Equations for Hypersurfaces

• Covariant differentiation in a submanifold of a Riemannian manifold
• The second fundamental form, the Gauss formulas, and Gauss' equation; Synge's inequality
• The Weingarten equations and the Codazzi–Mainardi equations for hypersurfaces
• The classical tensor analysis description
• The moving frame description
• Addendum: Auto-parallel and totally geodesic submanifolds
• Problems

Chapter 2. Elements of the Theory of Surfaces

• The first and second fundamental forms
• Classification of points on a surface; the osculating paraboloid and the Dupin indicatrix
• Principal directions and curvatures, asymptotic directions, flat points and umbilics; all-umbilic surfaces
• The classical Gauss formulas, Weingarten equations, Gauss equation, and Codazzi–Mainardi equations
• Fundamental theorem of surface theory
• The third fundamental form
• Convex surfaces; Hadamard's theorem
• The fundamental equations via moving frames
• Review of Lie groups
• Application of Lie groups to surface theory; the fundamental equations and the structural equations of SO(3)
• Affine surface theory; the osculating paraboloids and the affine invariant conformal structure
• The special affine first fundamental form
• Quadratic and cubic forms; apolarity
• The affine normal direction; the special affine normal
• The special affine Gauss formulas and special affine second fundamental form
• The Pick invariant; surfaces with Pick invariant 0
• The special affine Weingarten formulas
• The special affine Codazzi–Mainardi equations; the fundamental theorem of special affine surface theory
• Problems

Chapter 3. A Compendium of Surfaces

• Basic calculations
• The classical flat surfaces
• Ruled surfaces
• Quadric surfaces
• Surfaces of revolution
  • Rotation surfaces of constant curvature
• Minimal surfaces
• Addendum: Envelopes of 1-parameter families of planes
• Problems

Chapter 4. Curves on Surfaces

• Normal and geodesic curvature
• The Darboux frame; geodesic torsion
• Laguerre's theorem
• General properties of lines of curvature, asymptotic curves, and geodesics
• The Beltrami–Enneper theorem
• Lines of curvature and Dupin's theorem
• Conformal maps of ℝ³; Liouville's theorem
• Geodesics and Clairaut's theorem
• Addendum: Special parameter curves
• Addendum: Singularities of line fields
• Problems

Chapter 5. Complete Surfaces of Constant Curvature

• Hilbert's lemma; complete surfaces of constant curvature K > 0
• Analysis of flat surfaces; the classical classification of developable surfaces
• Complete flat surfaces
• Complete surfaces of constant curvature K < 0

Chapter 6. The Gauss–Bonnet Theorem and Related Topics

• The connection form for an orthonormal moving frame on a surface; the change in angle under parallel translation
• The integral of K dA over a polygonal region
• The Gauss–Bonnet theorem; consequences
• Total absolute curvature of surfaces
• Surfaces of minimal total absolute curvature
• Total curvature of curves; Fenchel's theorem, and the Fary–Milnor theorem
• Addendum: Compact surfaces with constant negative curvature
• Addendum: The degree of the normal map
• Problems

Mini-Bibliography for Volume III · Notation Index · Index

The Other Volumes

Tables of contents: Vol. 1 · Vol. 2 · Vol. 4 · Vol. 5. See also the Differential Geometry collection and the biography and complete bibliography of Michael Spivak.