Stephen Lovett’s book, Differential Geometry of Manifolds, a sequel to Differential Geometry of Curves and Surfaces, which Lovett co-authored with Thomas Banchoff, looks to be the right book at the right time. With the sixth and last chapter of the book under review being tantalizingly titled, “Applications of manifolds to physics,” the recent breakthroughs in algebraic topology having to do with Perelman’s proof of the Poincaré conjecture and the effect of Thurston’s geometrization program immediately spring to mind.
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. Indeed, one speculates that the indicated two-volume set, taking the reader from curves in R3, i.e. multivariable calculus, to some pretty cool hypermodern applications (Ch. 6 deals with, among other things, string theory and general relativity) explicitly aims to follow this novel trail.
Differential Geometry of Manifolds is also quite user-friendly (which, in my opinion as a non-geometer, is a relative rarity) in the sense that, for instance, Riemann does not meet Christoffel anywhere in its pages. No, Lovett doesn’t pitch as hard as he might: “this book attempts to provide an introduction to differentiable manifolds… geared toward advanced undergraduate or beginning graduate readers and retaining a view toward applications in physics.” Fair enough.
Lovett also notes that his book “can serve well either as a textbook or for self study,” and I completely agree. If my department ever lets me teach the advanced undergraduate course in differential geometry Lovett’s book s will feature heavily in the game. I suppose the book under review is best used for a second semester course, given that its predecessor, Banchoff-Lovett, focuses on curves and surfaces first, before, we get to manifolds per se. Lovett’s pacing is obviously not only pedagogically sound, it is kind.
It is additionally worth stressing that the pedagogical perspective adopted in Differential Geometry of Manifolds scores on yet another front. The book starts with a thorough review of advanced calculus with proper attention paid to the inverse and implicit function theorems. It is only after over thirty pages of such “warm-up” that we get to differential geometry proper. Again, this is a user-friendly book.
After his first chapter review Lovett starts the game with style. The second section of his second chapter, “Coordinates, frames, and tensor notation,” deals with the subject of moving frames in physics, and the chapter ends with a long section titled, “Tensor notation.” Lovett notes, tellingly, that “[m]athematicians and physicists often present tensors and the tensor product in very different ways, sometimes making it difficult for the reader to see that authors in different fields are talking about the same thing.” Truer words were never spoken, and, given that I belong to a working group including a physicist who talks a lot about tensors as vectors-gone-wrong (or is it the other way round?), I intend to use this section of Differential Geometry of Manifolds to great personal advantage myself.
And it is indeed the case that this feature of forging a bridge between generally warring dialects is characteristic of the book. For instance, Chapter 5, § 4, “The curvature tensor,” is another illustration of Lovett’s way of bringing about détente between mathematicians and physicists. After a pretty sporty (but altogether accessible) discussion of such themes as the Ricci and Bianchi identities, we encounter, happily, a section titled, “Geometric interpretation,” including, for instance, the following revelation: “…intuitively speaking, the torsion tensor gives a local measure of how much parallel transport of two non-collinear directions with respect to each other fails to close a parallelogram…” Very cool.
This section also contains a good discussion of the notion of an Einstein metric (which occurs, by definition, when the metric tensor and Ricci curvature are proportional) and that of an Einstein space (a manifold with an Einstein metric). And yes, we are on the verge of general relativity (appearing 33 pages later), but Lovett is sure first to address the hot mathematical issue of manifold classification: “The Uniformization Theorem, a fundamental result in the theory of surfaces, establishes that every connected 2-manifold admits a Riemannian metric with constant Gaussian curvature. This… leads to a classification of diffeomorphism classes of surfaces… One could hope that. In parallel with surfaces, all connected higher-dimensional manifolds (dim M > 2) would possess an Einstein metric that would… lead to a classification theorem of diffeomorphism classes of manifolds. This turns out not to be the case. There … exist higher dimensional compact manifolds that admit no Einstein metric…” Tantalizing.
Differential Geometry of Manifolds also comes equipped with a lot of problems for the student, a lot of good examples, and three useful appendices: on point set topology, on calculus of variations (which is particularly appropriate in connection with the interface with physics, of course), and on multilinear algebra. Finally, the book’s margins are very wide: doodle away!
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 is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.