Customer Reviews

Differential Geometry (Dover Books on Mathematics)

5 star:		(2)
4 star:		(2)
3 star:		(1)
2 star:		(0)
1 star:		(0)

 

 

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.

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 gives a good introduction to the topic, although a good background in mathematics (analysis) is very helpful. While not one of the easiest books to read, it serves as a good supplemental text to anyone browsing through other texts on differential geometry.

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.