Customer Reviews

Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201)

5 star:		(3)
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3 star:		(1)
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This is probably the most clearly written self-contained book on the basics of differential geometry. The author does a great job explaining the ideas behind purely mathematical 'dry' constructions. On the other hand, everything is defined correctly and precisely. A very readable and useful book with the perfect combination of formal math. and intuition. I would recommend it to students in theoretical physics area, together with the Nakahara's fantastic book.

This text is phenomenally easy to read and well organized. The author starts you on a journey by first explaining the importance and power of classifying manifolds namely by certain invariants preserved by certain mappings ( diffeomorphisms ).

For example, like Euler, we could count the number of holes in the surface and using this combinatorial method we are led to homology theory.

Or like Gauss, we could use a differentiation and integration to come up with the idea of curvature as an intrinsic feature of the surface.

Modern approaches use differential forms to represent homology and cohomoly groups.

The author also deals with fibre bundles demonstrating their importance in analyzing manifolds specifically how the number of fibre bundles possible with given Lie groups as structure groups over the manifold can be answered by characteristic classes such as the Chern and Pontrjagin classes. The use of differential forms is indispensible.

Perhaps the most satisfying aspect of this book is that it clarifies the notions of connection, connection form, curvature, curvature form for manifolds and fibre bundles.

There are plenty of exercises to boot.

This is a wonderful book. It is an insightful and careful introduction to differential forms and to the geometry they describe. The author is properly rigorous in his approach, but is kind enough to incorporate some informal discussion that gives much improved guidance. So, I find this a very much better learning opportunity than Flanders or Cartan or even Lovelock and Rund. I think it is a very helpful balance between correctness and full regard of the formalism and insight. On the other hand, there is a flavor that I would want included in such a book. The "phone book" version of Gravitation, by Misner, Thorne, and Wheeler, offers pictorial guidance in places. An improved version of that style of guidance would, I think, make this a perfect book. As it is, it would be absurd to criticize Misner, Thorne, and Wheeler. It has been a classic for years and will continue to be. It is well above my poor power to add or detract, for sure. Still, I find the phone book to be too loosely organized. It is encyclopaedic, but not crisp, insightful, and to the point. If just a bit of it could be incorporated here properly, we wouldn't need the phone book, because we'd already have the number.

This books gives a very direct explanation of the main concepts in differential forms. I would recommend it for anyone wanting to get to the main concepts quickly and cleanly.