(Continued from the review of Volume I.)
In the second volume, Spivak begins to study the classical parts of differential geometry. He does just the right thing: assuming the language and background developed in the first volume, he goes through the material on curves and surfaces that one typically meets in a first elementary course.
The curves portion is pretty much what one would expect. This material is so straightforward that the new language really doesn't add much.
Once one starts talking about surfaces, however, Spivak really gets interesting. He takes a quasi-historical approach, meaning that he follows the historical order of progress when it is helpful and ignores it when it is not. After a brief chapter on the theory of surfaces before Gauss, what we get is a careful reading of Gauss's famous Disquisitiones Circa Superficies Curvas. (One of the innovations in this third edition is that it includes the full text, with Spivak's comments on facing pages. Hooray!) Jumping off from Gauss, the chapter reworks the theory in modern terms.
Then he does it again: starting from Riemann's famous essay "On the Hypotheses which lie at the Foundations of Geometry," Spivak develops the basic ideas in the geometry of manifolds, including a brilliant section called "The Birth of the Riemann Curvature Tensor." This is crucial: if one doesn't see how the curvature tensor is the natural way to generalize the notion of curvature, then all is lost.
The chapters that follow set up the rest of the basic structure of modern differential geometry. The subsequent volumes then apply all this: see the tables of contents of volumes three, four, and five (which the author says should be considered one multivolume book) for what is covered.
Throughout all this, Spivak maintains a sense of humor and a clear head. Chapter 10 in Volume five is called "And now a brief message from our sponsor," for example. It deals with partial differential equations. (I have heard rumors about yellow pigs, but cannot verify them.)
The books' presentation is charming: the covers are paintings in a primitive style that include references (some clear, some subtle) to each volume's mathematical content. (The joke about the Klein bottle from the 2nd edition is, alas, gone.) The fifth volume, appropriately, has a banner on the cover saying "All the way with Gauss-Bonnet." Sounds right to me.
Certainly, going "all the way" was an amazing achievement for the author, and any reader who manages to follow him through all five volumes will learn a lot, both about differential geometry and about the value of diligence. Few readers will do that, but many will read parts of this book, and those who do will profit from it.
There is simply no book comparable to this one, so no library should be without this five volume set. At the price, individuals may well want a set themselves, or at least the first two volumes. The books can be ordered from http://www.mathpop.com.
Fernando Q. Gouvêa is professor of mathematics at Colby College.