Spivak, Differential Geometry Vol. 4, 3rd Edition — Table of Contents

The complete table of contents of A Comprehensive Introduction to Differential Geometry, Volume IV, 3rd Edition, by Michael Spivak (Publish or Perish, Inc.; ISBN 9780914098737), covering submanifolds in higher dimensions and codimensions, constant curvature manifolds, the second variation and comparison theorems, and variations of length, area and volume including minimal surfaces. Chapter and section headings are listed as they appear in the book.

Contents of Volume IV

Chapter 1. Higher Dimensions and Codimensions

• The geometry of constant curvature manifolds
  • The standard models of Sⁿ(K₀) and Hⁿ(K₀) in ℝⁿ⁺¹
  • Stereographic projection and the conformal model of Hⁿ
  • Conformal maps of ℝⁿ and the isometries of Hⁿ
  • Totally geodesic submanifolds and geodesic spheres of Hⁿ
  • Horospheres and equidistant hypersurfaces
  • Geodesic mappings; the projective model of Hⁿ; Beltrami's theorem
• Curves in a Riemannian manifold
  • Frenet frames and curvatures
  • Curves whose jth curvatures vanish
• The fundamental equations for submanifolds
  • The normal connection and the Weingarten equations
  • Second fundamental forms and normal fundamental forms; the Codazzi–Mainardi equations
  • The Ricci equations
  • The fundamental theorem for submanifolds of Euclidean space
  • The fundamental theorem for submanifolds of constant curvature manifolds
• First consequences
  • The curvatures of a hypersurface; Theorema Egregium; formula for the Gaussian curvature
  • The mean curvature normal; umbilics; all-umbilic submanifolds of Euclidean space
  • All-umbilic submanifolds of constant curvature manifolds
  • Positive curvature and convexity
• Further results
  • Flat ruled surfaces in ℝᵐ
  • Flat ruled surfaces in constant curvature manifolds
  • Curves on hypersurfaces
• Complete surfaces of constant curvature
  • Modifications of results for surfaces in ℝ³
  • Surfaces of constant curvature in S³: surfaces with constant curvature 0; the Hopf map
  • Surfaces of constant curvature in H³: Jörgens' theorem; surfaces of constant curvature 0; surfaces of constant curvature −1; rotation surfaces of constant curvature between −1 and 0
• Hypersurfaces of constant curvature in higher dimensions
  • Hypersurfaces of constant curvature in dimensions > 3
  • The Ricci tensor; Einstein spaces; hypersurfaces which are Einstein spaces
  • Hypersurfaces of the same constant curvature as the ambient manifold
• Addendum: The Laplacian
• Addendum: The ∗ operator and the Laplacian on forms; Hodge's Theorem
• Addendum: When are two Riemannian manifolds isometric?
• Addendum: Better imbedding invariants
• Problems

Chapter 2. The Second Variation

• Two-parameter variations; the second variation formula
• Jacobi fields; conjugate points
• Minimizing and non-minimizing geodesics
• The Hadamard–Cartan Theorem
• The Sturm Comparison Theorem; Bonnet's Theorem
• Generalizations to higher dimensions; the Morse–Schoenberg Comparison Theorem; Myers' Theorem; the Rauch Comparison Theorem
• Synge's lemma; Synge's Theorem
• Cut points; Klingenberg's theorem
• Problems

Chapter 3. Variations of Length, Area, and Volume

• Variation of area for normal variations of surfaces in ℝ³; minimal surfaces
• Isothermal coordinates on minimal surfaces; Bernstein's Theorem
• Weierstrass–Enneper representation
• Associated minimal surfaces; Schwarz's Theorem
• Change of orientation; Henneberg's minimal surface
• Classical calculus of variations in n dimensions
• Variation of volume formula
• Isoperimetric problems
• Addendum: Isothermal coordinates
• Addendum: Immersed spheres with constant mean curvature
• Addendum: Imbedded surfaces with constant mean curvature
• Addendum: The second variation of volume

Mini-Bibliography for Volume IV · Notation Index · Index

The Other Volumes

Tables of contents: Vol. 1 · Vol. 2 · Vol. 3 · Vol. 5. See also the Differential Geometry collection and the biography and complete bibliography of Michael Spivak.