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Differential Geometry of Manifolds [Hardcover]

Stephen T. Lovett

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Book Description

June 11 2010 1568814577 978-1568814575

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.


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Review

Differential Geometry of Curves and Surfaces and Differential Geometry of Manifolds will certainly be very useful for many students. A distinguishing feature of the books is that many of the basic notions, properties and results are illustrated by a great number of examples and figures. Each section includes numerous interesting exercises, which make these books ideal for self-study too. These books give a nice addition to the existing literature in the field of differential geometry of curves, surfaces, and manifolds. I strongly recommend them to anyone wishing to enter into the beautiful world of the differential geometry.
—Velichka Milousheva, Journal of Geometry and Symmetry in Physics, 2012

It provides a broad introduction to the field of differentiable and Riemannian manifolds, tying together the classical and modern formulations. … The book takes a practical approach, containing extensive exercises and focusing on applications of differential geometry in physics…
L’Enseignement Mathématique (2) 57 (2011)

Lovett fills with this book a blatant gap in the vast collection of books on differential geometry. The book is easily accessible for students with a basic understanding of partial derivatives and a basic knowledge of vector spaces. …
it provides a thorough understanding of the most important concepts and thus opens the way for further studies, either in differential geometry (many references to other textbooks that go deeper into the subjects are included in the book) or in other research areas where differential geometry provides the language and tools to describe and solve the area’s problems. An ample number of examples and exercises stimulate mastery in handling the tools introduced in the text.
The book is well suited for an introductory course in differential geometry, graduate students in mathematics or other sciences (physics, engineering, biology) who need to master the differential geometry of manifolds as a tool, or any mathematician who likes to read an inspiring book on the basic concepts of differential geometry.
—G. Paul Peters, Mathematical Reviews, Issue 2011k

Driven by the desire to generalize multivariate analysis to manifolds, the author guides the reader through the concepts of differential manifolds, their tangent spaces, vector fields, and differential forms and their integration. … The last chapter distinguishes this book from others in the field. It features applications of the mathematical theory to physics … Throughout the book, the introduction of a new notion is clearly motivated, relations to the classical theory are established, and notational conventions are explained. … This works very well … the book is self-contained to a high degree and suitable as textbook for a lecture or for self-study.
—H.-P. Schröcker, International Mathematical News, August 2011

Intended to provide a working understanding of the differential geometry of n-dimensional manifolds, it does a good deal more, offering treatments of analysis on manifolds (including the generalized Stokes’s theorem) in addition to Riemannian geometry. An especially interesting chapter on applications to physics includes some general relativity, string theory, symplectic geometry, and Hamiltonian mechanics. … Highly recommended.
—S.J. Colley, CHOICE, February 2011

… the right book at the right time. … We live in an age when borders between mathematical disciplines (and even between parts of mathematics and parts of physics) are being re-drawn — or erased altogether — and differential geometry is a major player in all this Perestroika. Thus, the teaching of the subject to rookies should perhaps be restructured, too, at least in the sense of getting to the more avant garde stuff more quickly, and it looks like a major aim of Lovett's book is exactly that. … I think this is going to be a very successful textbook especially for rookie graduate students (and the zealous undergraduate would-be differential geometer, of course), as well as a very popular self-study source. It is a very nice book indeed.
—Michael Berg, Loyola Marymount University, Los Angeles, California, USA

About the Author

Stephen Lovett is an associate professor of mathematics at Wheaton College in Illinois. Lovett has also taught at Eastern Nazarene College and has taught introductory courses on differential geometry for many years. Lovett has traveled extensively and has given many talks over the past several years on differential and algebraic geometry, as well as cryptography.

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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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Amazon.com: 3.0 out of 5 stars  2 reviews
24 of 24 people found the following review helpful
4.0 out of 5 stars Clear, concise, rigorous introduction but possibly difficult for self-study Oct. 16 2010
By gengogakusha - Published on Amazon.com
Format:Hardcover|Verified Purchase
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.
2.0 out of 5 stars Not for self study Dec 27 2013
By V. F. Ho - Published on Amazon.com
Format:Hardcover|Verified Purchase
I was looking for a self-study introductory book on DG and manifolds, to strengthen my basis for General Relativity study, but this book is not what I was looking for.

Starting somewhere in Chapter 3 I was not able to follow 100% of the material. The "definition+theorem+proof" methodology might be good for rigorousness, but is terrible as pedagogy, and it is not conducive to building your geometrical intuition. I was looking for a book that explains the motivation behind a given (and usually strange) definition, instead of using the "fallen from the gods of the Olympus" approach.

The book relies on the other book by the author more than what I expected.

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