Differential Geometry

Texte du rabat

This is a text of local differential geometry considered as an application of advanced calculus and linear algebra. The discussion is designed for advanced undergraduate or beginning graduate study, and presumes of readers only a fair knowledge of matrix algebra and of advanced calculus of functions of several real variables.
The author, who is a Professor of Mathematics at the Polytechnic Institute of New York, begins with a discussion of plane geometry and then treats the local theory of Lie groups and transformation groups, solid differential geometry, and Riemannian geometry, leading to a general theory of connections.
The author presents a full development of the Erlangen Program in the foundations of geometry as used by Elie Cartan as a basis of modern differential geometry; the book can serve as an introduction to the methods of E. Cartan. The theory is applied to give a complete development of affine differential geometry in two and three dimensions.
Although the text deals only with local problems (except for global problems that can be treated by methods of advanced calculus), the definitions have been formulated so as to be applicable to modern global differential geometry. The algebraic development of tensors is equally accessible to physicists and to pure mathematicians. The wealth of specific resutls and the replacement of most tensor calculations by linear algebra makes the book attractive to users of mathematics in other disciplines.

Contenu

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