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.

Table of Contents for 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

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