One of my all-time favorite books is J. J. Rotman’s *Advanced Modern Algebra*, not just because I love algebra and have done so since my freshman year at the university (courtesy of McCoy and Herstein), but because whenever I had to learn things beyond what I later learned in graduate school, Rotman’s book was perfect: it’s all there and it’s presented beautifully. Of course, this should be understood modulo the fact that I am not an algebraist, but a fellow traveler, in my case, a number theorist: expert algebraists (like Rotman himself) would presumably go elsewhere; Rotman’s book aims at a broad(ish) audience.

So, why bring up algebra when the book under review is about differential geometry? Well, my claim is that Lee’s *Introduction to Smooth Manifolds* is very similar to Rotman’s book in the hugely beneficial effect it exercises: I have over recent years had (and certainly still have) occasion to work with manifolds of different flavors, and I am ecstatic to have Lee’s book in my possession. I fully expect to use it in much the same way as Rotman’s book!

But I think there may be another *caveat* to be taken note of. Just as Rotman’s book is not meant for a novice, so Lee’s book (as a Springer GTM entry, in point of fact) presupposes a bit on the part of its reader: “general topology, the fundamental group,… covering spaces, [and] basic undergraduate linear algebra and real analysis.” Thus, to be sure, Lee’s description of his work as “a first-year graduate text on manifold theory” is entirely on target; however, it’s worth noting that he has also “tried to focus on portions of manifold theory that will be needed by most people who go on to use manifolds in mathematical or scientific research.”

This underscores the fact that the book is ideally poised as a second book in a sequence, the first entry being something like Munkres’ by-now classic *Topology*, or Lee’s own *Introduction to Topological Manifolds*, or, given that some familiarity with Riemannian manifolds is useful, the last two chapters in what this Lee refers to as a book by “Jeffrey Lee (no relation),” i.e., the latter’s AMS GSM entry, *Manifolds and Differential Geometry*, or the present Lee’s book, *Riemannian Manifolds: an Introduction to Curvature*. Lee cites several other books along these lines, in order to indicate that his audience should certainly not contain raw recruits.

This said, however, much like Rotman’s algebra text, this *Introduction *covers a lot of material in a largely self-contained fashion. Take the critical notion of Riemannian metric, for example. On pp. 327–328 we read that “A Riemannian metric … is a smooth symmetric covariant 2-tensor field on [a smooth manifold] *M* that is positive definite at each point.” He goes on to say that “[i]f *g* is a Riemannian metric on *M*, then for each [point] *p* [on] M, the 2-tensor *g*_{p} is an inner product on [the tangent space] *T*_{p}M … [and] because of this we often use the notation <*v*,*w*>_{g} to denote the real number *g*_{p}(*v*,*w*) …”

Note that this is often presented in the opposite order in more elementary texts: one starts with the metric as a smoothly varying inner product and then realizes its tensor properties in virtue of its bilinearity, etc. Shortly after this Lee presents the Euclidean metric as the simplest example of a Riemannian metric (the coefficient functions are Kronecker deltas) and then races to a very crisp proof of the fact that Riemannian metrics exist for all smooth manifolds. I recently came across much of this material in Rosenberg’s *The Laplacian on a Riemannian Manifold*, which takes a much more pragmatic approach with the goal being Hodge-de Rham theory: Lee’s discussion is beautifully complementary, adding a lot of detail to the structure and bringing the geometry and topology into the foreground. The point is that, to be sure, Lee expects something from his readers in the way of mathematical maturity, but he provides huge dividends. And his presentation is at the same time very systematic and clear, and he makes sure that it’s all treated in a very thorough and complete (if concise) manner. Again, a perfect approach for the level he aims at.

*Introduction to Smooth Manifolds *is a big book, of course (as is Rotman’s), coming in at around 700 pages. Its contents are properly predictable, but at times surprising: all the i’s are dotted and all the t’s are crossed, and Lee pushes the reader to some more *avant* *garde* stuff (consider e.g. the book’s last chapter, on symplectic manifolds). It starts off with five chapters covering basics on smooth manifolds up to submersions, immersions, embeddings, and of course submanifolds. Lee hits Sard’s theorem before getting to Lie theory and vector fields.

Thereafter (we’re now in Chapter 9) he goes on to integral curves, flows, bundles, and tensors, and then the stage is set for Riemannian geometry proper in Chapter 13. Chapters 14–18 cover differential forms and de Rham theory, with Chapter 18 devoted to de Rham’s critically important theorem. Finally, Lee turns to distributions and foliations, more Lie theory, quotient manifolds, and (as mentioned above) symplectic manifolds. This sketch of the book’s contents suffices to underscore that indeed, as already indicated, Lee seeks to bring the reader to familiarity with “the portions of manifold theory that will be needed by most people who go on to use manifolds in ... research”; certainly, the emphasis on Hodge-de Rham theory and the inclusion of symplectic manifolds in the line-up testify to this objective.

By the way, the way Lee treats Hodge’s theorem itself is illustrative of the fact that the book isn’t pitched too high either: the Laplace operator occurs in an exercise on p. 464, the Hodge *-operator having been defined in another exercise some 25 pages earlier. Lee then has the reader do a number of instructive sub-problems about harmonic forms, sufficient to get to the statement of Hodge’s theorem in a meaningful fashion, and then properly refers to more specialized sources for the proof.

So, yes, the book under review is laden with excellent exercises that significantly further the reader’s understanding of the material, and Lee takes great pains to motivate everything well all the way through: see, for instance, his discussion (in Chapter 16) of integration on manifolds, replete with an abundance of good “drawings” (or pictures, I guess).

I learned what differential geometry I know from such books as Spivak’s *Calculus on Manifolds*, Tu’s *An Introduction to Manifolds*, and the aforementioned book by Rosenberg. Lee’s book forms a marvelous umbrella for this material, as well as Riemannian geometry proper, and then goes quite a bit further: it’s a fine graduate-level text for differential geometers as well as people like me, fellow travelers who always wish they knew more about such a beautiful subject.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.