Multivariable Calculus and Differential Geometry

The back cover of this book sports a picture of the author (unfortunately small). Gerard Walschap appears equipped with a cigarette dangling from his lip, proceeding to pour beer into a Klein bottle. A picture of this quality really should be featured on the cover of the book, and let the chips fall where they may. There is so little humor in the academic world today, with pre-tenure neurosis a pandemic among the youths and cartels of clones vying angrily for a piece of the pie, that Walschap’s “in your face” self-portrait is more than welcome. I am reminded of Jack Nicholson’s remark to Shirley MacLaine in Terms of Endearment to the effect that she needs “a lot of drinks” — perhaps this prescription would work well for our current mathematical species, too. Walschap has the right idea, the Klein bottle’s unforgiving nature notwithstanding.

With such a bit of artwork on the back cover, the contents of the book had better be good, too. I am happy to be able to report that they are. Walschap sets himself the task of introducing “the reader to the basic concepts of differential geometry with a minimum of prerequisites,” and then goes on to say that modulo some mathematical maturity (referred to as “helpful”), “[t]he only absolute requirement is a solid background in single variable calculus.”

This is wonderful, I think: generally the student, i.e. the advanced undergraduate, headed into a differential geometry course is supposed to have multivariable calculus under his belt, but this is generally a dicey business. For one thing, there is the difficulty of getting to the main theme in modern differential geometry, manifolds, in a pretty expeditious manner. Going, say, the do Carmo route (his Differential Geometry of Curves and Surfaces is a standard text at this level) is all good and well, but the difficulty here is that covering just curves properly is a time-consuming project; adding surfaces to the bill makes for a full course, but in a way it is all a postponement of the study of abstract manifolds. The same can be said for, e.g., Banchoff and Lovett whose (excellent) book carries the same title as do Carmo’s. At the moment I am using do Carmo in a tutorial and I have used Banchoff-Lovett in a full semester upper division course. To be sure, these are good books. However, I find it to be a great problem that it’s such a slow ascent to the generalities of n-manifolds, and for this reason alone I would recommend Walschap’s book.

I guess it’s fair to say that in some ways Walschap’s approach reminds me of Spivak’s in Calculus on Manifolds, with the important qualification that Walschap goes much more slowly and his book is therefore much more user-friendly. Additionally, Walschap is very thorough and is careful not only to spell things out in great detail (his audience being composed of neophytes, after all) but to pepper the text with a huge number of examples. He also provides a great number of good exercise sets: they are long and graduated in the sense that the problems sweep out a spectrum from routine calculations to more serious (but still accessible) proofs. So pedagogically speaking this is an excellent book indeed, particularly for the intended level.

I am most happy, however, with the layout of the book. Walschap finishes his two opening chapters on, respectively, Euclidean spaces and differentiation, in a little over 100 pages, capping off the latter chapter with a discussion of the incomparably important topics of vector fields, Lie brackets (yes, we have derivations knocking on the door, but that’s for a later course), and partitions of unity. Next, curves are properly covered in chapter two, as are the inverse and implicit function theorems (shades of Spivak’s approach: this important analytic material should come early), but no time is wasted to get to manifolds in chapter three. And now the games begin in earnest: this chapter includes coverage of the tangent bundle, covariant differentiation, geodesics, and curvature (and more besides), all of them topics which are ordinarily saved for a graduate level course in differential geometry, I think.

Walschap does a good job, too, in playing both ends against the middle as far as his audience’s mathematical youthfulness is concerned: he focuses on manifolds situated in an ambient Euclidean space and he spends plenty of time en passant, as it were, on the case of surfaces in 3-space. The idea is, as he puts it, to capitalize on the fact that “it is much more intuitive to consider a surface in 3-space and make the leap of imagination to higher dimensions than to study an abstract topological space…” This, again, is evidence of the high pedagogical quality of this work.

Chapters four and five are de rigeur: they deal with integration on Euclidean spaces and subsequently differential forms and integration on manifolds. Stokes’ theorem appears in sections 5.6 and 5.7. The fifth chapter finishes with a discussion of closed forms and exact forms — no de Rham cohomology yet, but that’s how it should be, since undergraduates are, well, undergraduates. Down the line the kids going on to graduate work should look at (one of my favorite early graduate school books:) Loring Tu’s An Introduction to Manifolds and subsequently the seminal text, Differential Forms in Algebraic Topology, by Bott and Tu — it doesn’t get any better than that.

To get back to business, the last two chapters of Walschap’s book concern manifolds as metric spaces (with the sixth chapter including the Hopf-Rinow theorem — the only other place I’ve come across this result is in Milnor’s Morse Theory, where it is covered toward the end of his “Rapid course in Riemannian geometry”) and, finally, hyperspaces. The book closes with an appendix on the analysis situs of the real numbers and one on linear algebra.

I like this book a great deal. It is ambitious in the sense that Walschap tries to get a lot of serious mathematics done with an audience with very little preparation, but I think his approach succeeds. If the student pays close attention to this book, and stays with it (and works hard, of course), the payoff is substantial. The kid will learn not only a great deal of differential geometry, but other allied material besides, particularly analysis in general finite dimensional Euclidean spaces. Again, it is a very sound pedagogical effort and deserves success.

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