Customer Reviews

A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition

5 star:		(7)
4 star:		(1)
3 star:		(1)
2 star:		(0)
1 star:		(1)

 

 

This book is the first volume of the 3rd edition in a five volume series on differential geometry. The emphasis on this first volume is the study of differential forms and de Rham Cohomology Theory. Spivak also considers two 'bonus' topics: integral manifolds & foliations and Lie groups.

You'll need some prerequisites to get started. For the differential topology material (including Sard's Theorem and Whitney's 2n+1 Embedding Theorem), I recommend Hirsch's Differential Topology. For results on determinants and symmetric groups, I use Hungerford's Algebra, now in its 12th printing. For the general topology material (Hausdorff spaces, Urysohn metrization, etc.), I recommend Munkres' Topology (2nd Edition).

Spivak begins this volume with a review of topological manifolds in Chapter 1. The author provides the basic definitions and gives lots of examples of surfaces and other manifolds. The discussion of manifolds and surfaces continues in the Chapter 1 Exercises. (The author routinely used the exercise set to continue the thread of discussion.) Quick mention of the surface classification theorem is made, although for the proof of this, you'll need to look in Hirsch or Munkres. The reader gets to have fun gluing topological handles onto and cutting disks out of the 2-sphere.

Chapter 2 reviews some of the basic concepts from differential topology, including the fundamental Whitney Embedding Theorem and Sard Critical Point Theorem. Basic properties of smooth maps are also studied.

Chapter 3 studies the general vector bundle and specializes to the tangent bundle of a smooth manifold. The author is keen on the idea that the reader 'grok' (i.e. understand intuitively) the tangent bundle and the associated induced maps and commutative diagrams. The notion of orientability is also introduced.

Multilinear forms and their tensor product are studied in Chapter 4. This is a key building block in the construction of de Rham cohomology. The author gets side tracked a bit with a discussion of differences in classical/modern notion.

Chapter 5 is a very nice chapter on vector fields. Instead of just appealing to results from differential equations (as is usually done) to build integral curves and the flow of a vector field, Spivak establishes these needed results from differential equations using a very accessible integral equations/fixed point argument. Once the flow of a vector field is show to exist (locally), Lie derivatives and Lie brackets are then studied.

Following the integral curves & vector fields material in the previous chapter, the author detours a bit and studies the problem of integral manifolds of dimensions other than 1 along with applications to foliations in Chapter 6. Spivak establishes a basic version of the Frobenius Integrability Theorem and uses examples to motivate the result before diving into the proof.

The basics of de Rham cohomology are established in Chapter 7 and Chapter 8. Alternating and skew-symmetric forms are discussed, although is may be easiest to establish some of the needed results on the symmetric group of permutations after reviewing Hungerford's Algebra. Differential forms and their wedge product are defined, and Frobenius' Theorem can now be restated in terms of differential forms. Two versions of Stokes Theorem are established and this result is applied to integrating forms on manifolds and studying properties of the degree of a proper map of between manifolds. The formal definition of the de Rham cohomology groups is given and some basic calculations are carried out.

The author does something curious with one of the main results of de Rham cohomology, namely the homotopy-invariance property. He starts this with a discussion section in Chapter 7 (not a called out theorem) in which contractible manifolds are show to have zero cohomology in all dimension by an explicit calculation showing all closed k-forms are exact. The results that the author establishes in Chapter 7 for this `one-off' calculation are precisely what are needed to show the more general result that homotopic maps induce equivalent homomorphisms of de Rham cohomology later in Chapter 8.

Chapter 9 is a very nice chapter covering several foundational topics of Riemannian geometry; include the Riemannian metric, geodesics, the exponential map, geodesic completeness and tubular neighborhoods.

Chapter 10 is a short chapter on Lie groups and is something of a detour from the main thread. The author uses the material as a source of application of the material from the first nine chapters.

Returning to de Rham cohomology in Chapter 11, more foundational results from algebraic topology are studied, including exact sequences, Poincare Duality, the Thom class and the index of a vector field.

The book contains many wonderful geometric diagrams which help motivate the material. In most cases, the author is very careful to highlight theorems, propositions and lemmas. Occasionally key results will be 'buried' in a series of discussion paragraphs, which makes referring to these results later on somewhat difficult. The author never, ever calls out or highlights any of his definitions. This can be somewhat frustrating, especially when trying to track down one of these definitions. Fortunately the index to the book is reasonably good.

Michael Spivak begins these five volumes stating his modest aim to write the "Great American Differential Geometry book." He surely has. Instead of listing the numerous subjects Spivak treats clearly and beautifully in these volumes, I'd like to call out the delightful travelogue style in which they are written, using history, anecdotes, and opinion to explain, illuminate, and, when possible, motivate the gleaming modern edifice. Spivak's opinions are sprinkled lightly here and there like easter eggs. How could you not love a math book that uses the subtitle "The Debauch of Indices," or dismisses Eric Temple Bell's history as "supercilious remarks of questionable taste"? Also, don't miss the annotated bibliography in volume 5. The fact that legions of professionals refer to these books in their original *typewritten* format [1st & 2d editions] is a further testament to their quality. The third edition is typeset using TeX and, though beautiful, still manages to retain a little of the quirky typewritten appearance. One quibble: I was disappointed to see that this edition did not use Richard Bassein's bizarre artwork [think 70s psychedelic] for the covers; I admit that this stuff weirded me out originally, but have grown to love it -- where else could I see fuzzy trolls in crowns made from Enneper's minimal surface?

Let Spivak take you "All the Way With Gauss-Bonnet."

Spivak's text gets a lot of good reviews, and it is a fine text. In fact, it's one of the best I've ever seen. Read a few other books on the subject, and you'll agree that this is a massive improvement on them. So why only 3 stars? Because there's a much better text on the subject: John Lee's "An Introduction to Smooth Manifolds". This book outshines Spivak's in so many ways. Sure, Spivak is great at motivating major developments in the theory (for instance, he really helps you understand why we need to define a tangent space and why it is the way it is), but he fails pretty bad when it comes to developing some actual theory.

Reading Spivak's text is like taking a stroll, a fresh break from the usual mathematics textbook style. But you also hit a bunch of brick walls on this stroll. It'll be a great discussion, and then you'll come to a theorem. You'll have no idea what its for (some of the time) and you'll struggle to work through its proof (most of the time). Furthermore, the organization is... well, there is no organization! As a result, Spivak can seem to droll on. Lee isn't as good at giving the overall big picture as well as Spivak, but he does everything else exceptionally. Leave Spivak for bed time reading, but do your real studying out of Lee.

Volume I of Spivak's five volume series is a very good introduction to differential geometry. The strength of Spivak's book is that it's not just a series of theorems and examples, but is full of explanations. For example some books define a manifold to be a set covered by the charts of an atlas, others require a manifold to have a maximal atlas. Spivak doesn't just give one of these definitions -- he explains why you might prefer one to the other. Another example: he gives three different definitions of the tangent space, shows they are equivalent, and then explains why they turn out to be equivalent.

In comparison to another very good introduction, do Carmo's "Differential Geometry of Curves and Surfaces", Spivak is more abstract. Spivak works with vector bundles and tensors, do Carmo does everything without these concepts (although he does have a brief discusion of the tangent bundle at the end of the book). do Carmo's book covers a lot of topics that don't appear until later volumes of Spivak, such as global differential geometry, the Gauss-Bonnet theorem and the theorem egregium. On the other hand, Spivak Volume I has a chapter on Lie Groups, which aren't discussed in do Carmo.

Spivak's book has a lot of good exercises. It also has some misprints, but they are easily corrected. I didn't find any significant errors. In particular I found Theorem 7 on pp. 205-207 to be clearly explained.

If you are looking for an introductory book on differential geometry you should very seriously consider Spivak.

If you want a book that is rich with examples then this is it. The proofs are, for the most part, clear and concise, thus a person who is learning the material without the aid of an instructor can follow the logic. However, the author could have spent some more time developing topological ideas (thought he does have an appendix section that does a fair job of it) within the flow of the first chapter. I personally find appendices to be too distracting and tend to slow down the flow of the material in a particular chapter. Other than that, this is a great book if you want to learn differential geometry and the theory of smooth manifolds.

A lively, terribly ambitious tome on differential geometry. It was meant as a guided tour through the jungles of geometry, from a historical perspective. It is neither easy to read nor altogether successful in it's aim, but it IS comprehensive, masterful, and absolutely unlike all the others. It's kind of a legend since virtually every mathematician seems to own a copy. Full of pictures and history. Reads like a novel.

I see we have 3rd edition now. I got 1st.

SO, I got it while it was hot!

So many pages that some folks will be put off. Exquisite writing however. See also his elementary book called Calculus. Bingo, you'll learn to appreciate him!

Spivak's explanation to hard stuff is awesome. Sometimes daily chat language is used, he gave satisfactory and very good examples. I like it.

You can find cites about Spivak's Differential Geometry in most books about it. Also in a lot scientific papers. It have all you must know about Differential Geometry and more. Every aspect is covered and explained in detail. You have to read it.

Those who recommend it haven't used this book. I used to revere the five volumes. Then I started to read it. And the exposition doesn't hold water!

Ex.1: Read the proof of Theorem 7 on pp. 205-207. Is the proof correct? Not really.

Beginners, don't waste your time on this book.