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The Geometry of Physics: An Introduction

Theodore Frankel
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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I often tell students how important differential geometry is for understanding modern physics. Of course, I'm not an expert in either field, just a fan. As a fan, my knowledge of the literature is limited. That said, Theodore Frankel's The Geometry of Physics looks pretty good to me.

The table of contents of Frankel's book is impressive: it starts with manifolds and differential forms, progresses to discuss integration and Stokes' theorem, introduces the Lie derivative and holonomy. At that point we're around page 190 or so. But it's a huge book, with a lot more to come, covering most of differential geometry, from the geometry of surfaces in three-dimensional space all the way to Lie groups, vector bundles, topological quantization, covering spaces, and homotopy groups. While doing all that mathematics, physical applications are kept in mind and introduced as necessary. So, for example, the chapter on vector bundles ends with a discussion of the Electromagnetic Connection. Minkowski space and pseudo-Riemannian manifolds (whose metrics are not necessarily positive definite) are kept in view, since they are crucial in Relativity Theory. In fact, Frankel has posted a list of questions in physics and engineering that are considered in the book, probably to convince physics students that learning the geometry is worth the effort.

I suspect that mathematics students would also find this book congenial. Frankel adds to every section title a question that captures the main issue at stake. For example, section 4.1, on "The Lie Derivative of a Vector Field" has the question "Walk one mile east, then north, then west, then south. Have you really returned?" Exactly. And how could anyone resist a section title such as "Geodesics, Spiders, and the Universe". (The question for that one is "Is our space flat?")

Of course, treating all this material, even in a book with almost 700 pages, requires quite a bit of compression. This isn't bed-time reading. On the other hand, it's also not a formal mathematics book. People who like to read that kind of book will probably find Frankel wordy and insufficiently rigorous. I imagine that Frankel's style will appeal to students who are willing to accept a less than completely formal account and to professionals, who can supply the formalism in most cases and really need to hear the core ideas, the "what's really going on" information that's usually only shared orally. If you're looking for a well-written and well-motivated introduction to differential geometry, this one looks hard to beat.

Fernando Q. Gouvêa's research interests are in the history of mathematics and in number theory. He is a big fan of good expository writing in mathematics.

Preface; Part I. Manifolds, Tensors and Exterior Forms: 1. Manifolds and vector fields; 2. Tensors and exterior forms; 3. Integration of differential forms; 4. The Lie derivative; 5. The Poincar‚ lemma and potentials; 6. Holonomic and non-holonomic constraints; Part II. Geometry and Topology: 7. R3 and Minkowski space; 8. The geometry of surfaces in R3; 9. Covariant differentiation and curvature; 10. Geodesics; 11. Relativity, tensors, and curvature; 12. Curvature and topology: Synge's theorem; 13. Betti numbers and de Rham's theorem; 14. Harmonic forms; Part III. Lie Groups, Bundles and Chern Forms: 15. Lie groups; 16. Vector bundles in geometry and physics; 17. Fiber bundles, Gauss-Bonnet, and topological quantization; 18. Connections and associated bundles; 19. The Dirac equation; 20. Yang-Mills fields; 21. Betti numbers and covering spaces; 22. Chern forms and homotopy groups; Appendix A. Forms in continuum mechanics; Appendix B. Harmonic chains and Kirchhoff's circuit laws; Appendix C. Symmetries, quarks, and meson masses; Appendix D. Representations and hyperelastic bodies; Appendix E: Orbits and Morse-Bott theory in compact Lie groups.