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-Stieltjes 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