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Differential Geometry: Bundles, Connections, Metrics and Curvature

Clifford Henry Taubes
Oxford University Press
Publication Date: 
Number of Pages: 
Oxford Graduate Texts in Mathematics 23
[Reviewed by
Michael Berg
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Taubes’s preface to the book under review makes my job as reviewer trivial, at least inasmuch the details of “what”, “for whom,” and “how” are concerned. The “what” is answered by Taubes’s comment that the book “is meant to be an introduction to the subject of vector bundles, principal bundles, metrics (Riemannian and otherwise), covariant derivatives, connections and curvature.” Then the “for whom” is addressed by: “I am imagining an audience of first year graduate students or advanced undergraduate students who have some familiarity with the basics of linear algebra and with the notion of smooth manifold.” Finally, the “how” is taken care of by such lines as “I have tried to make the presentation as much as possible self-contained with proofs of basic results presented in full …” and “I have worked out many examples in the text, because truth be told, the subject is not interesting to me in the abstract … I need to feel [Taubes’s emphasis] the geometry to understand what is going on …”

Now that I’ve let Taubes himself do much of my work for me, let me atone and make a few additional observations. First, Taubes ends the Preface with noting that he himself learnt the subject from Raoul Bott in “his first-semester graduate differential geometry class,” and this makes for (if you’ll pardon the puns) a number of fabulous connections. There is no doubt that one of the best sources for algebraic topology with an orientation toward differential geometry, if not the very best, is the book, Differential Topology in Algebraic Topology, by Bott and Tu (to which I would append as a prelude, An Introduction to Manifolds, by Tu), and therefore Taubes’s present book marvelously rounds out this chunk of higher mathematical education.

Second, Taubes’s style is very fetching. Every chapter is preceded by a paragraph or two (or even more) leading the reader gently into the particular part of the forest that lies ahead: a very nice touch. Here are three samples.

  • (Ch. 1) “… differential geometry studies various constructions that can be built on smooth manifolds … my favorite [textbook on differential topology is] Differential Topology by Victor Guillemin and Alan Pollock …”
  • (Ch. 4) “Any linear operations that can be done to generate a new vector space from some given initial set of vector spaces can be done fiber-wise with an analogous set of vector bundles to generate a new vector bundle …”
  • (Ch. 10) “A principal bundle is the Lie group analog of a vector bundle, and they are [sic] in any event, intimately related to vector bundles.”

These introductions tend to go far to give the reader a sense, before the fact, of what the purpose and stresses are of what he is about to encounter. This is clearly a very good pedagogical ploy.

Speaking of pedagogy, I should like to note, in closing, three more pluses and one pseudo-minus. On the plus side, (1) Taubes’s proofs are wonderful, complete and elegant (modulo the comments in his Preface: as far as some of the preliminaries go he is content to refer to other sources — fair enough). (2) He is clear and concise, and indeed presents all the material beautifully and in a self contained manner, to the degree possible. (3) He amplifies everything with generous allusions to other sources so that the reader is easily able to follow up on certain themes with considerable ease. This approach considerably mitigates the single quibble I have, i.e. (“–1”) Taubes presents no explicit sets of exercises to the reader, although he does provide a number of examples. On the other hand, via the many, many fine references in the book this reader can easily find other sources for problems and exercises. Additionally, if this same, presumably zealous, reader seeks to get his hands nice and dirty, he will be more than satisfied by the experience of working through the book by “arguing” with the text in its margins, which is, of course, the way any good mathematics text should be read.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

1. Smooth manifolds
2. Matrices and Lie groups
3. Introduction to vector bundles
4. Algebra of vector bundles
5. Maps and vector bundles
6. Vector bundles with fiber Cn
7. Metrics on vector bundles
8. Geodesics
9. Properties of geodesics
10. Principal bundles
11. Covariant derivatives and connections
12. Covariant derivatives, connections and curvature
13. Flat connections and holonomy
14. Curvature polynomials and characteristic classes
15. Covariant derivatives and metrics
16. The Riemann curvature tensor
17. Complex manifolds
18. Holomorphic submanifolds, holomorphic sections and curvature
19. The Hodge star
Indexed list of propositions by subject