*Manifolds, Tensors, and Forms.* Right off it’s about differential geometry. Excellent. But then we get to the subtitle: “An Introduction for Mathematicians and Physicists.” The title itself already hints at the problem: we mathematicians certainly know a lot about manifolds and (differential) forms, seeing that these are mainstays of any serious differential geometry course, particularly at the beginning graduate level. It’s the middle word that causes trouble: we know that a tensor product is obtained by means of an equivalence relation on pairs of vectors — the main move is to factor out by bilinearity. More generally, factor out by mutilinearity, depending on how many vector spaces are in the game at once. Additionally we might be playing with *R*-, *k*-, or *G*-modules (yes: rings, fields, groups acting: fabulous algebraic fun! For me it was love at first sight, many, many moons ago.)

But then there’s the physicists’ way of doing things: I remember in my early high school days when I was at my most Poindexter, reading about Einstein’s relativity in some popular account (possibly Lincoln Barnett’s pithy *The Universe and Dr. Einstein*) and coming across something denoted by *R*_{ik} which was called, yes, a tensor. What’s the story, then? What does *R*_{ik} have to do with bilinearity? Well, at that point I thought the word “tensor” was some cool and mysterious physics term, but presently, with my quickly losing interest in physics proper and finding my *dharma* (as Weil puts it in his truly wonderful autobiography, *The Apprenticeship of a Mathematician*) in pure mathematics, I came to feel that what physicists thought about things was irrelevant — it all left me totally cold, 0 K, in fact. Who cares what *R*_{ik} might have to do with bilinearity?

Well, little did I know: there’s no small element of irony in the fact that now, over forty years later, my own pure mathematics involves a huge amount of stuff the physicists have done, and are doing, and the chickens have come home to roost: physics-speak is unavoidable. So, once again, what does *R*_{ik} have to do with bilinearity? And here’s the answer as given by Dirac in his *General Theory of Relativity*, on p. 7: “… *T*^{α′β′}_{γ′}=x^{α′}_{,λ}x^{β′}_{,μ}x^{ν}_{,γ′}T^{λμ} . Any quantity that transforms according to this law is a tensor.” So there. Here Dirac is using the Einstein summation convention, meaning that on the right we’re actually summing over *λ* and *μ*, and we have that the *T*’s are linear operators acting on suitable functions of *x*, so, bilinearity begins to emerge, even if a bit *sub rosa*.

Who was it who compared mathematicians and physicists to Englishmen and Americans — peoples separated by a common language? To be sure!

So what does all this have to do with the book under review? Well, let’s see what Renteln says about tensors. His second chapter (p. 30, ff.) starts with the observation that “mathematicians introduce tensors formally as a quotient of a certain module, while physicists introduce tensors using objects with many indices that transform in a specific way under a change of basis.” A very promising start, indeed! He goes on to say that “[w]e follow a middle approach here.” Fair enough. Then he starts with the generic object *v*⊗*w*, with *v* and *w* being ordinary vectors, and goes on to form the algebra of all tensors (of all orders) using the moves we mathematicians are all familiar with, and, sure enough, working in degree 2, say, we get the formula *T*=Σ_{ij}T^{ij}ei⊗*e*_{j} for the most general second-order tensor on **R**^{3} (with the *T*^{ij} scalars, of course). So it is that now bilinearity is worked into the inner life of *v*⊗*w* or, basis-wise, the *e*_{i}⊗*e*_{j}_{,} and we see crystal-clearly why this subject is called linear algebra — just insist that everything in sight behave linearly. Renteln makes all this explicit in a particularly interesting way: he builds up the roster of players step-by-step, getting to multigrading rather swiftly: “if *S* is a tensor of type (*r*,*s*) and *T* is a tensor of type (*p*,*q*), then *S*⊗*T* is a tensor of type (*r*+*p*, *s*+*q*)” (recall that “type (*r*,*s*) means that we’re playing with *V*^{⊗}^{r}⊗*V**^{⊗}^{s}), and only then hits multilinearity with full gusto. It works well.

In fact, the whole book works well. It is written at a relatively slow pace (even as Renteln advertises that his style is terse so as to facilitate self-study) and is replete with lots of examples, historical musings, sets of exercises, and of course discussions of how things tie in with physics. Nonetheless, it covers a good deal of ground, taking the reader from the foundational stuff in linear and multilinear algebra and differentiation on manifolds (so more or less what you’d get in Spivak’s *Calculus on Manifolds*) to rather serious heights. The sequence of more beefy chapters (the main course of the book, so to speak), is as follows: homotopy plus de Rham cohomology; homology; integration on manifolds (Spivak does some of this, too, of course, in *loc. cit.*); vector bundles (including the method of moving frames); geometric manifolds (including Riemann’s curvature tensor — which leads to the Ricci curvature tensor, the erstwhile *R*_{ik} , on p. 206. This eighth chapter ends with a section on Hodge theory); and finally Renteln hits some true topology: his last chapter, “The degree of a smooth map,” does the hairy ball theorem, Hopf fibration, linking numbers (and magnetostatics!), and Poincaré-Hopf plus Gauss-Bonnet. Quite a cornucopia. I like the book a great deal.

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