Differential Geometry

Designed for advanced undergraduate or beginning graduate study, this text contains an elementary introduction to continuous groups and differential invariants; an extensive treatment of groups of motions in euclidean, affine, and riemannian geometry; and development of the method of integral formulas for global differential geometry.

Preface
Chapter 1. Elementary Differential Geometry
  1-1 Curves
  1-2 Vector and Matrix Functions
  1-3 Some Formulas
Chapter 2. Curvature
  2-1 Arc Length
  2-2 The Moving Frame
  2-3 The Circle of Curvature
Chapter 3. Evolutes and Involutes
  3-1 The Riemann-Stieltjès Integral
  3-2 Involutes and Evolutes
  3-3 Spiral Arcs
  3-4 Congruence and Homothety
  3-5 The Moving Plane
Chapter 4. Calculus of Variations
  4-1 Euler Equations
  4-2 The Isoperimetric Problem
Chapter 5. Introduction to Transformation Groups
  5-1 Translations and Rotations
  5-2 Affine Transformations
Chapter 6. Lie Group Germs
  6-1 Lie Group Germs and Lie Algebras
  6-2 The Adjoint Representation
  6-3 One-parameter Subgroups
Chapter 7. Transformation Groups
  7-1 Transformation Groups
  7-2 Invariants
  7-3 Affine Differential Geometry
Chapter 8. Space Curves
  8-1 Space Curves in Euclidean Geometry
  8-2 Ruled Surfaces
  8-3 Space Curves in Affine Geometry
Chapter 9. Tensors
  9-1 Dual Spaces
  9-2 The Tensor Product
  9-3 Exterior Calculus
  9-4 Manifolds and Tensor Fields
Chapter 10. Surfaces
  10-1 Curvatures
  10-2 Examples
  10-3 Integration Theory
  10-4 Mappings and Deformations
  10-5 Closed Surfaces
  10-6 Line Congruences
Chapter 11. Inner Geometry of Surfaces
  11-1 Geodesics
  11-2 Clifford-Klein Surfaces
  11-3 The Bonnet Formula
Chapter 12. Affine Geometry of Surfaces
  12-1 Frenet Formulas
  12-2 Special Surfaces
  12-3 Curves on a Surface
Chapter 13. Riemannian Geometry
  13-1 Parallelism and Curvature
  13-2 Geodesics
  13-3 Subspaces
  13-4 Groups of Motions
  13-5 Integral Theorems
Chapter 14. Connections
Answers to Selected Exercises
Index