Differential Geometry of Manifolds

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From the coauthor of Differential Geometry of Curves and Surfaces , this companion book presents the extension of differential geometry from curves and surfaces to manifolds in general. It provides a broad introduction to the field of differentiable and Riemannian manifolds, tying together the classical and modern formulations. The three appendices provide background information on point set topology, calculus of variations, and multilinear algebra� topics that may not have been covered in the prerequisite courses of multivariable calculus and linear algebra. Differential Geometry of Manifolds takes a practical approach, containing extensive exercises and focusing on applications of differential geometry in physics, including the Hamiltonian formulation of dynamics (with a view toward symplectic manifolds), the tensorial formulation of electromagnetism, some string theory, and some fundamental concepts in general relativity.

Differential Geometry of Manifolds Review

Lovett provides a nice introduction to the differential geometry of manifolds that is useful for those interested in physics applications, including relativity. It is clearly written, rigorous, concise yet with the exception of the complaints mentioned below, generally reader-friendly and useful for self-study. The difficulty level is midway between O'Neill's Elementary Differential Geometry, Revised 2nd Edition, Second Edition and Tu's An Introduction to Manifolds (Universitext Volume 0).

The pace is quite fast. As you can see in more detail from the "search inside this book" function:

Ch. 1 Analysis of Multivariable Functions [pp. 1-36] provides some background math;

Ch. 2 [pp. 37-78] Coordinates, Frames, and Tensor Notation discusses some more applied topics needed for physics applications;

Ch. 3 Differential Manifolds [pp. 79-124] and Ch. 4 Analysis on Manifolds [pp. 125-184] discuss essential standard topics including differential maps; immersions, submersions and submanifolds; vector bundles; differential forms; integration and Stokes' Theorem;

Ch. 5 [pp. 185-248] provides an introduction to Riemannian Geometry, including vector fields, geodesics and the curvature tensor; and finally

Ch. 6 [pp. 249-294] provides very brief discussions of some applications to physics including Hamiltonian mechanics, electromagnetism, string theory and general relativity.

I like the fact that it includes an exposition of Pseudo-Riemannian metrics in section 5.1.4 and 5.3.3 and in section 6.4, a short introduction to general relativity. It's the only book I am familiar with that can help one make the leap from very elementary books like O'Neill's Elementary Differential Geometry, Revised 2nd Edition, Second Edition, Pressley's Elementary Differential Geometry (Springer Undergraduate Mathematics Series) or Banchof and Lovett's Differential Geometry of Curves and Surfaces to graduate level books like Tu's An Introduction to Manifolds (Universitext Volume 0), John Lee's Introduction to Smooth Manifolds, Jeffrey Lee's massive Manifolds and Differential Geometry (Graduate Studies in Mathematics) or for the relativity buffs, O'Neill's brilliant Semi-Riemannian Geometry With Applications to Relativity, 103, Volume 103 (Pure and Applied Mathematics), all of which I also recommend after Lovett.

Now for the drawbacks:

(1) My main gripe is that there are no answers to problems, which detracts from its value for self-study (but to fill that gap, cf. Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers). This is especially annoying because Lovett refers to answers to some problems in his mathematical exposition, e.g., on p. 234 (section 5.4.1), he refers to problem 5.2.17 on page 217 in his discussion of connections that are not symmetric; moreover answers to some exercises depend on material in other problems, e.g., the answer to problem 5.2.17 refers to problem 5.2.14. This is an all too common practice I dislike because it seriously degrades from a book's value for self-study. Overall, this is a small part of the book.

(2) A Heads up: some of the exposition in Ch 5 Introduction to Riemannian Geometry strikes me as a bit too terse and the demands on one's stamina and ability to comprehend highly abstract mathematical concepts is highest. One example you can partly check out for yourself with Search Inside is section 5.2 Connections and Covariant Differentiation (cf. pp 204-206). If you're ok with that, you should be "good to go". This is too bad as this is chapter is fascinating and the material is required for a modern understanding of relativity.

(3) A minor point to be aware of is that the physics applications are extremely terse. To be fair to Lovett he does state in the preface that he does not "supply all the physical theory". Fair enough.

Overall, this textbook is a useful addition to the many books on differential geometry because of its refreshing, "no nonsense" clarity, rigor and conciseness as well as the topics covered. It seems to me suitable for self-study provided you are confident in your math skills, have the required prerequisites and can tolerate the fact that in some places, the development rests on results you are expected to provide without any guidance. Since I read more than one book on a subject as a matter of course, these drawbacks / limitations were not a show-stopper for me but they might be for others.

UPDATE 10/29/2011: Lovett perhaps deserves only a *** 1/2 star rating based on the drawbacks I mentioned. I rounded up because I consider *** stars a mediocre evaluation and I do think the book has merit. I'd rely more on my description than the stars.

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