Spivak, Differential Geometry Vol. 2, 3rd Edition — Table of Contents

The complete table of contents of A Comprehensive Introduction to Differential Geometry, Volume II, 3rd Edition, by Michael Spivak (Publish or Perish, Inc.; ISBN 9780914098713), covering the classical theory of curves and surfaces, Gauss's and Riemann's original work in translation, and the many faces of the curvature tensor from the Ricci calculus to connections in principal bundles. Chapter and section headings are listed as they appear in the book.

Contents of Volume II

Chapter 1. Curves in the Plane and in Space

• Curvature of plane curves
• Convex curves
• Curvature and torsion of space curves
• The Serret–Frenet formulas
• The natural form on a Lie group
• Classification of plane curves under the group of special affine motions
• Classification of curves in ℝⁿ

Chapter 2. What They Knew About Surfaces Before Gauss

• Euler's Theorem
• Meusnier's Theorem

Chapter 3. The Curvature of Surfaces in Space

• How to read Gauss
• Gauss's theory of surfaces
  • The Gauss map
  • Gaussian curvature
  • The Weingarten map; the first and second fundamental forms
  • The Theorema Egregium
  • Geodesics on a surface
  • The metric in geodesic polar coordinates
  • The integral of the curvature over a geodesic triangle
  • Addendum: The formula of Bertrand and Puiseux; Diquet's formula

Chapter 3A. Gauss's Disquisitiones generales circa superficies curvas — a translation

Chapter 4. The Curvature of Higher Dimensional Manifolds

• An inaugural lecture
  • "On the Hypotheses which lie at the Foundations of Geometry"
• What did Riemann say?
  • The form of the metric in Riemannian normal coordinates
• A prize essay
• The birth of the Riemann curvature tensor
  • Necessary conditions for a metric to be flat
  • The Riemann curvature tensor
  • Sectional curvature
  • The Test Case; first version
  • Addendum: Finsler metrics

Chapter 5. The Absolute Differential Calculus (the Ricci Calculus); or, the Debauch of Indices

• Covariant derivatives
• Ricci's Lemma
• Ricci's identities
• The curvature tensor
• The Test Case; second version
• Classical connections
• The torsion tensor
• Geodesics
• Bianchi's identities

Chapter 6. The ∇ Operator

• Koszul connections
• Covariant derivatives
• Parallel translation
• The torsion tensor
• The Levi-Civita connection
• The curvature tensor
• The Test Case; third version
• Bianchi's identities
• Geodesics
• The First Variation Formula
• Addendum: Connections with the same geodesics
• Addendum: Riemann's invariant definition of the curvature tensor

Chapter 7. The Repère Mobile (the Moving Frame)

• Moving frames
• The structural equations of Euclidean space
• The structural equations of a Riemannian manifold
• The Test Case; fourth version
• Adapted frames
• The structural equations in polar coordinates
• The Test Case; fifth version
• The Test Case; sixth version
• "The curvature determines the metric"
• The 2-dimensional case
• Cartan connections
• Covariant derivatives and the torsion and curvature tensors
• Bianchi's identities
• Addendum: Manifolds of constant curvature
  • Schur's Theorem
  • The form of the metric in normal coordinates
• Addendum: Conformally equivalent manifolds
• Addendum: É. Cartan's treatment of normal coordinates

Chapter 8. Connections in Principal Bundles

• Principal bundles
• Lie groups acting on manifolds
• A new definition of Cartan connections
• Ehresmann connections
• Lifts
• Parallel translation and covariant derivatives
• The covariant differential and the curvature form
• The dual form and the torsion form
• The structural equations
• The torsion and curvature tensors
• The Test Case; seventh version
• Bianchi's identities
• Summary
• Addendum: The tangent bundle of F(M)
• Addendum: Complete connections
• Addendum: Connections in vector bundles
• Addendum: Flat connections

Notation Index · Index

The Other Volumes

Tables of contents: Vol. 1 · Vol. 3 · Vol. 4 · Vol. 5. See also the Differential Geometry collection and the biography and complete bibliography of Michael Spivak.