An Introduction to Differential Manifolds [Paperback]

Dennis Barden (Author), Charles B. Thomas (Author)

Book Description

Publication Date: March 2003 | ISBN-10: 1860943551 | ISBN-13: 978-1860943553

An introduction to basic ideas in differential topology, based on the many years of teaching experience of both authors. Among the topics covered are smooth manifolds and maps, the structure of the tangent bundle and its associates, the calculation of real cohomology groups using differential forms (de Rham theory), and applications such as the Poincare-Hopf theorem relating the Euler number of a manifold and the index of a vector field. Each chapter contains exercises of varying difficulty for which solutions are provided. Special features include examples drawn from geometric manifolds in dimension 3 and Brieskorn varieties in dimensions 5 and 7, as well as detailed calculations for the cohomology groups of spheres and tori.

Editorial Reviews

Review

"This book is an excellent introductory text into the theory of differential manifolds with a carefully thought out and tested structure and a sufficient supply of exercises and their solutions. It does not only guide the reader gently into the depths of the theory of differential manifolds but also careful on giving advice how one can place the information in the right context. It is certainly written in the best traditions of great Cambridge mathematics." Acta Scientiarum Mathematicarum

From the Publisher

Readership: Upper level undergraduates, beginning graduate students, and lecturers in geometry and topology --This text refers to an out of print or unavailable edition of this title.

Product Details

• Paperback: 232 pages
• Publisher: Imperial College Press (March 2003)
• Language: English
• ISBN-10: 1860943551
• ISBN-13: 978-1860943553
• Product Dimensions: 8.9 x 6.3 x 0.4 inches
• Shipping Weight: 11.2 ounces (View shipping rates and policies)
• Average Customer Review: 4.5 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #1,376,752 in Books (See Top 100 in Books)

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8 of 9 people found the following review helpful:

4.0 out of 5 stars A mile wide and a yard deep, March 4, 2009

By 

Malcolm (Tokyo) - See all my reviews

This review is from: An Introduction to Differential Manifolds (Paperback)

Barden & Thomas's "Introduction to Differential Manifolds" has the broadest coverage of any introductory graduate text in differential topology that I've seen, even more than Lee's Introduction to Smooth Manifolds or Guillemin & Pollack's Differential Topology, and in less than 200 pages. Not only does it cover the standard topics found in all such books, i.e., the rank theorem, diffeomorphisms, immersions, embeddings, tangent bundles, Sard's theorem, the Whitney embedding theorem, etc.; more topological topics, such as degree theory, the Poincare-Hopf theorem, Morse theory, and handlebodies; and the usual material with a more analytic or geometric flavor, such as differential forms, tensors, vector bundles, integration, Stokes's theorem, de Rham cohomology, and Lie groups and algebras, but there is also a chapter on fibre bundles (which already is rare for a book of this level) that includes further material on classifications of higher-dimensional manifolds that probably has never appeared in book-form before. Unfortunately, the treatment of many of these topics is rather cursory, with the most interesting material being the least well explained.

Except for Chapter 3 and the latter parts of Chapter 7, the book should be accessible for first-year graduate students. The explanations are a little more detailed than those of, say, Broecker & Jaenich's Introduction to Differential Topology, although not to the point of the spoon-feeding ones finds in Lee or Guillemin & Pollack, so students with no prior exposure to manifold topology may find that it moves a little too fast (all the material mentioned above fits into the first 165 pages). There's a 20-page refresher appendix on differential analysis, covering the prerequisites for the book, such as metric spaces, Banach spaces, (an unhelpful definition of) tensor products, the inverse function theorem, Sard's theorem, etc., although (1) much of it is too brief to help you if you don't know it already, (2) not all of it is really necessary for this book (e.g., don't worry if you don't know what a Banach space is), and (3) the full proof (modulo some mistakes) of Sard's theorem is given, so you don't need to have learned it elsewhere. There are between 2 and 8 exercises at the end of each chapter (relatively few when compared to other introductory texts), whose level ranges from the routine to the very difficult, but this is because they are "intended for students to work at and then discuss with a supervisor." However, "for the benefit of readers working independently," a chapter of solutions for the exercises is included, which is big plus.

The main problem with the book is that it tries to do too much, as there is no topic here that is not covered better in some other book, even though no other book covers all this material. Some of the chapters are way too brief, with the most egregious examples being that on Lie groups and Lie algebras (12 pages to cover both, including maximal tori and cohomologies of compact Lie groups), de Rham cohomology (it would really help if the reader has some exposure to algebraic topology first), and Morse theory and handle decompositions. One byproduct of squeezing in too many topics is that the proofs start to become rushed in the last couple of chapters, with lots of handwaving and some mistakes, as the care and precision in definitions and notation of the earlier chapters evaporates. Some of the biggest mistakes/weaknesses in this insufficiently copyedited book include: an overly restrictive proof of the Whitney embedding theorem; a mix-up in the proof of Stokes's theorem between equations on the manifold and those in the local coordinate neighborhood; at least 6 mathematical typos in the equations for Lemma 5.5.1; and 6 serious errors in the proof that every manifold admits Morse functions, including omitting the word "degenerate" in front of "critical point" and twice stating the exact opposite of what is intended(!). Furthermore, on p. 155 the authors state an important technical result on Lie subgroups and then prove only a special case to avoid dealing with the full technicalities, but implicit in their proof of the special case is actually an unrecognized use of the general case.

However, probably the worst mistakes occur on pp. 143-6 in an attempted proof of the Morse homology theorem that includes "open" sets that are not open, a Mayer-Vietoris cover that fails to cover the manifold in question (and a good thing, too, because if it did the formula would be wrong), and a diagram (Fig. 7.6) that clearly does not match the text. (You will have to learn Morse theory and handle decompositions from a different book, such as Hirsch's Differential Topology or Kosinski's Differential Manifolds, although even elementary books such as Gauld's Differential Topology: An Introduction or Wallace's Differential Topology: First Steps are better.) As a final example of a botched proof, I mention Sard's theorem, which the authors commendably attempted to state and prove for the minimal differentiability required. Unfortunately, what they ended up doing was stating the theorem properly in the case where the domain and range spaces have equal dimension, but then making a mistake in the estimation of the constants in the proof (cf. de Rham's proof, in Varietes differentiables, which they attempted to imitate, but missed), and misstating the theorem in the case where the dimensions are unequal, but then presenting the correct (except for a tiny mix-up) proof (from Milnor's Topology from the Differentiable Viewpoint) for the smooth case. See Sternberg's Lectures on Differential Geometry for a proper proof of the full theorem.

On the other hand, a few things are handled particularly well in the book: The various (5) definitions for tangent vectors (although I prefer Broecker & Jaenich's treatment, which these authors cite as an inspiration for this book), the proofs of the Poincare lemma and Mayer-Vietoris theorem in the chapter on cohomology, and degree theory (namely, degrees of maps, indices of vector fields, winding numbers, the Euler characteristic, and the Poincare-Hopf theorem that relates them; similar to treatments in Milnor and Guillemin & Pollack). The most startling feature of this book is the additional topics it covers; in particular, in Chapter 3 on fibre bundles, there's a section on "applications" that presents Thurston's Geometrization Conjecture (or should I say theorem now?) for 3-dimensional manifolds, along with a motivating analogy from the familiar 2-dimensional situation and a partial explanation of some of the terms involved (such as "pseudo-Anosov"), although most of this discussion will be way over the readers' heads, and there is understandably no mention of the method of proof. Even more surprising is what follows: A statement of Dennis Barden's own classification theorem for simply connected, closed, smooth 5-manifolds, which to my knowledge has not appeared outside his original papers. Unfortunately, the theorem is only then related to some Brieskorn varieties, about which is given not much more than a definition and an example. This material should really have been reserved for a more advanced text.

6 of 29 people found the following review helpful:

5.0 out of 5 stars This book Rules!, May 29, 2003

By 

raymund aglipay (Monterey Park, CA United States) - See all my reviews

This review is from: An Introduction to Differential Manifolds (Hardcover)

This book is just so full of useful information and details. It has a lot of problems for which most of the solutions are supplied. Man, I love differential manifolds after spending some quality time with this book.