Geometry of Differential Forms

by Shigeyuki Morita

Since the times of Gauss, Riemann, and Poincaré, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms. The book by Morita is a comprehensive introduction to differential forms. It begins with a quick introduction to the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results concerning them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated is a detailed description of the Chern-Weil theory. The book can serve as a textbook for undergraduate students and for graduate students in geometry.

Genres: Mathematics, Geometry

321 pages, Paperback

First published September 1, 2001

Reviews

[Brian Powell · Rating 3.5 out of 5]

I came across this text while putting together a self-study curriculum to learn about characteristic classes. It has a good coverage of topics, starting at the basics with manifold theory, differential forms, and de Rham theory; these subjects occupy the first half of the book. The writing is clear and crisp, but terse. This is OK because these should be familiar topics to most readers. The latter half develops fiber bundles and characteristic classes (Pontryagin, Chern, and Euler; no Stiefel-Whitney) and I think here things get dicey. One becomes soon aware that this book actually is not at all pedagogical, and isn't really *teaching* you anything. It's a fairly traditional collection of definitions, theorems, and proofs, with virtually no elaboration, interpretation, or discussion. More mature readers might be fine here, but then I suppose they should already know this stuff.

While struggling through this text, I turned to Naber's two-volume set on gauge theory: his 2nd volume ends with a deep and lively chapter on characteristic classes, heavily motivated by applications to physics. It is excellent, and I think an apt place to first see some of this stuff. As for next steps, I'm not sure. I'm gearing up for Bott & Tu but the internet says it's going to kick my ass. That may be, we'll see. But I closed Morita about 80% through and I don't think I'll be revisiting it again.