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Publisher:

Dover Publications

Publication Date:

2007

Number of Pages:

336

Format:

Paperback

Price:

19.95

ISBN:

9780486457826

Category:

Monograph

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

Fernando Q. Gouvêa

09/29/2007

This is a Dover reprint of a book first published by Academic Press in 1980. J. Arthur Seebach's *Telegraphic Review* appeared in the January 1981 issue of the *American Mathematical Monthly*:

This text, based on a graduate course given at the University of Minnesota, gives a coherent introductory view of those aspects of differential geometry which are showing up in applications. In particular, the reader is introduced to Morse theory, Lie groups, dynamical systems, and catastrophe theory. 10 of the 11 chapters end with a short but substantial set of "problems and projects." Good bibliography, short index.

The book shows its age mostly in the fact that it concludes with a chapter on "singularities and catastrophes," then much in vogue. The style is quite formal but has flashes of charm, as when, in the first paragraph of chapter 1 (on "Manifolds and Their Maps"), the author says that

No political treaty has ever been drawn-up to delineate clearly between advanced calculus and the elementary theory of manifolds. I shall attempt, toward the end of this chapter, to give a concise presentation of this borderline material, hoping to be helpful to some without offending others.

I only wish that today's "advanced calculus" even got near to such sweet material as this!

Kahn's book provides a good introduction, aimed at graduate students, to a whole swath of interesting mathematics, including some topics that are not often included in such texts. I am happy that Dover has brought it back to print.

Fernando Q. Gouvêa is editor of MAA Reviews.

Preface

Introduction

1. Manifolds and Their Maps

2. Embeddings and Immersions of Manifolds

3. Critical Values, Sard's Theorem, and Transversality

4. Tangent Bundles, Vector Bundles, and Classification

5. Differentiation and Integration on Manifolds

6. Differential Operators on Manifolds

7. Infinite-Dimensional Manifolds

8. Morse Theory and Its Applications

9. Lie Groups

10. Dynamical Systems

11. A Description of Singularities and Catastrophes

Bibliography

Index

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