Differential Geomtery of Curves and Surfaces

Books on differential geometry can be classified as either “classical” (in which the object of study is curves and surfaces in the plane and space; see, for example, the books by Tapp or doCarmo) or “modern” (where manifolds and related objects take center stage; examples here include Tu’s Introduction to Manifolds or, if you’re ambitious, Spivak’s five-volume magnum opus). This book, however, is a hybrid; it starts with curves and surfaces but also contains an extended discussion of manifold theory.

Apart from appendices, there are three chapters. The first is on curves. This chapter discusses both the local (e.g., Frenet equations) and global (e.g., four vertex theorem) theory of curves in the plane and in space. Spiral curves are discussed in somewhat more detail than is customary, and there are also brief forays into such topics as homotopic curves, Möbius transformations and hyperbolic geometry.

One aspect of this chapter that I thought could be improved concerns the definition of the word “curve”. The authors never actually give one, at least not one specifically denominated “definition”. Instead they ease into the concept of a curve from an intuitive standpoint, drawing some figures then explaining that a curve can be viewed as the graph of a function or described parametrically. Ultimately they tend to focus on the parametric definition, but even here they depart from usual practice by viewing the curve as a set of points in the plane or space — i.e., as the image of a function, rather than the function itself. However, they also use notation (such as “the curve \(\gamma(s)\)”) that suggests the curve is the function rather than the graph, so it seems that making these kinds of hair-splitting distinctions is not a high priority with the authors.

The second chapter is on surfaces. The chapter begins, as did the previous chapter, with an intuitive overview, explaining various ways how surfaces can be defined — as the graph of a function, defined implicitly by an equation, or defined parametrically. After introducing surfaces, the chapter discusses such topics as the first and second fundamental forms, Gaussian and mean curvatures, geodesics, the Gauss-Bonnet theorem, and Gauss’s Theorema Egregium.

This classical theory is explored further in Appendix B of the text (Appendix A is a review of Euclidean spaces, multivariable calculus, and existence theorems for ordinary differential equations). Appendix B contains a smorgasbord (ten sections, each apparently independent of the others) of special topics in the theory of curves and surfaces, including, for example, evolutes, the isoperimetric inequality, cartography and the fundamental theorem of surface theory.

The first two chapters total about 130 pages; Appendix B adds another 65. There is probably more than enough material in these combined pages for a one-semester course on classical differential geometry, with prerequisites some calculus and linear algebra. The exposition is somewhat concise but generally clear, and there are a good number of diagrams provided to facilitate understanding. Despite the fact that this book is a translation from the Japanese, there are none of the problems with awkward English that sometimes accompany a translated work.

Each section ends with exercises, solutions to which appear in the back of the book. There are an adequate, if not overwhelmingly large, number of exercises, and by and large they tend to be nontrivial ones.

The book, however, contains more than this classical theory. The third and final chapter, which will almost certainly be beyond the reach of most undergraduates in this country, is on topics related to manifolds; the idea is to look at curves and surfaces from the perspective of manifold theory. After a discussion of differential forms, Riemannian manifolds are introduced, and results from chapter 2 (for example, the Gauss-Bonnet theorem) are looked at from this more general perspective.

This chapter is considerably more difficult than the rest of the book, and is addressed to an entirely different audience — one that is explicitly assumed to already know something about manifolds. (The authors give chapter 5 of Lecture Notes on Elementary Topology and Geometry by Singer and Thorpe as the kind of background that is assumed here.) However, it would seem that any student who already knows what a manifold is would not be reading the introductory chapters on curves and surfaces. The transition from classical curve and surface theory to this more sophisticated material in about sixty pages struck me as a somewhat jarring one.

I think the idea of providing a bridge from the classical to the modern theories of differential geometry is a very good one. However, this idea works best if the book actually develops the notion of a manifold and relates that concept to surfaces. This is the way it is done, for example, in chapter 7 of Elements of Differential Geometry by Millman and Parker. As done here, however, the “bridge” doesn’t quite go all the way from one shore to the other.

To summarize and conclude: in a class populated by students who already have some exposure to the concept of a manifold, the presence of chapter 3 in this text may make for an unusual and interesting course. Most classes, I suspect, will not be so populated, in which case the primary function of this book will be as a text for a more conventional course in the classical theory of curves and surfaces. While the book is certainly suitable for such a course, students may find the books by Tapp or Millman and Parker to be somewhat more accessible.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.