Differential Geometry (Dover Books on Mathematics) [Paperback]

Heinrich W. Guggenheimer (Author)

Book Description

Publication Date: June 1, 1977 | ISBN-10: 0486634337 | ISBN-13: 978-0486634333

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.

Product Details

• Paperback: 378 pages
• Publisher: Dover Publications (June 1, 1977)
• Language: English
• ISBN-10: 0486634337
• ISBN-13: 978-0486634333
• Product Dimensions: 8.2 x 5.6 x 0.8 inches
• Shipping Weight: 10.4 ounces (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #958,268 in Books (See Top 100 in Books)

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18 of 18 people found the following review helpful:

5.0 out of 5 stars Lots of math for the serious differential geometry student to chew on., November 13, 2006

By 

Daverz - See all my reviews

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This review is from: Differential Geometry (Dover Books on Mathematics) (Paperback)

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

17 of 19 people found the following review helpful:

5.0 out of 5 stars Not only for pure mathematician, September 5, 2000

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"ddfbike" (Napoli, ITALY EU) - See all my reviews

This review is from: Differential Geometry (Dover Books on Mathematics) (Paperback)

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.

6 of 7 people found the following review helpful:

4.0 out of 5 stars Don't the judge the book by the title, August 20, 2009

By 

R. Bagula "Roger L. Bagula" (Lakeside, Ca United States) - See all my reviews
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This review is from: Differential Geometry (Dover Books on Mathematics) (Paperback)

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.