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

Shigeyuki Morita (Author)

Book Description

Publication Date: August 28, 2001 | ISBN-10: 0821810456 | ISBN-13: 978-0821810453

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. <P>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. <P>The book can serve as a textbook for undergraduate students and for graduate students in geometry.

Editorial Reviews

Language Notes

Text: English (translation)
Original Language: Japanese

Product Details

• Paperback: 321 pages
• Publisher: American Mathematical Society (August 28, 2001)
• Language: English
• ISBN-10: 0821810456
• ISBN-13: 978-0821810453
• Product Dimensions: 8.5 x 5.5 x 0.8 inches
• Shipping Weight: 15.2 ounces (View shipping rates and policies)
• Average Customer Review: 4.5 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #222,185 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

37 of 37 people found the following review helpful:

5.0 out of 5 stars A very good book., March 28, 2005

By 

Gargantua (France) - See all my reviews

This review is from: Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201) (Paperback)

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.

20 of 20 people found the following review helpful:

5.0 out of 5 stars Self contained introduction to techniques of classifying manifolds., January 9, 2007

By 

Rehan Dost (Canada) - See all my reviews
(REAL NAME)   

This review is from: Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201) (Paperback)

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.

3.0 out of 5 stars Great Book, January 1, 2012

By 

Jim Curry (MT) - See all my reviews
(VINE VOICE)   

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This review is from: Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201) (Paperback)

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.