Differential Geometry

I had in the past bought another book with the same title from the same publisher (Dover books): Differential Geometry. They were even published first in the same year 1963 adding to confusion. This first book was sort of a standard text with very little imagination or in the long term worth. The book I'm reviewing in contrast gives tools for development and a catalog of surface types by their differential geometry. I think the most important thing is the development of the algebra and calculus of the affine geometry of surfaces. Some of the results here are very important in the study of general relativity and chaotic systems theory ( both). The difference between the two books is that the first I never go back to and this one I will spend some time trying to get more out of. I'm grateful to Heinrich W. Guggenheimer for writing this text: he gives me hope for mathematics.

Guggenheimer's book is a very solid introduction to differential geometry which emphasizes the Cartan moving-frame approach. This approach is used to produce invariants for surfaces under affine transformations, etc. Also included are some integral formulas which are used to show that spheres are the only star-shaped surfaces with constant mean curvature.

I think this must be the least expensive differential geometry book that uses Cartan's orthonormal frame method. Though more than 40 years old, the notation is essentially modern (there are a few typographical oddities which aren't really bothersome). This is a very rich book, with fascinating material on nearly every page. In fact, I think it's a bit too rich for beginners, who should probably start with a more focused text like Millman & Parker or Pressley. 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

I find the book very interesting: it's a very good presentation of "classical problems with modern methods" in Differential Geometry. It's appreciable for the selection of topics and their logical order, the clarity of their exposition (based on the use of modern terminology), the set of proposed problems and the relative results and the list of references at the end of each chapter.