A Differential Approach to Geometry: Geometric Trilogy III

With this volume, the author completes his trilogy of books covering the broad sweep and many facets of geometry. The first in the series (An Axiomatic Approach to Geometry) looked at geometry from a synthetic viewpoint, starting from its pre-Greek origins and working forward in time through the development of non-Euclidean geometry, finally culminating in a chapter giving a precise axiomatic development of the subject. The second (An Algebraic Approach to Geometry) used the methods of abstract and linear algebra to define and explore affine, Euclidean and projective geometry, and also provided an introduction to the theory of algebraic curves. In this book, the focus is on how calculus can be used to develop geometric ideas.

Like the other books in this trilogy, this one relies heavily on history as a motivational tool. The first chapter, in fact, constitutes a look at the evolution of definitions of things like curves and tangent lines; various definitions are proposed and examined, with problems inherent in them discussed. Eventually the author settles on definitions, and the work of studying (classical) differential geometry begins with chapter 2. Since chapter 1 uses history for motivational purposes, the rest of the book is essentially independent of it, and a course based on this book could certainly begin with chapter 2.

Curves are the subject of chapters 2 through 4. Plane curves are discussed in chapter 2 and space (or “skew”, as the author calls them) curves are the subject of chapter 4. These two chapters bookend a chapter devoted entirely to an examination of detailed examples — the deltoid, cissoid, tractrix, catenary, and more than a dozen others. Although the titles of these three chapters do not distinguish between the “local” and “global” theory, both aspects of the theory are discussed in them; for example, the Four-Vertex theorem, a global result, is proved. And of course the traditional topics of the local theory (e.g., the Frenet apparatus) are covered as well.

The remaining three chapters of the book deal with surfaces. Chapter 5, the first of them, discusses the classical local theory of surfaces in three-space. Topics include the definition of a surface, and the basic facts connected with them, including a heavy emphasis on curvature. In the next chapter, the notion of a surface is looked at from a different point of view (emphasizing its status as an entity in its own right rather than as a subset of three-space) to discuss its intrinsic properties, giving the author an opportunity to introduce the basic terminology and notions of Riemannian geometry in a nice concrete setting. (As a person who loathes the Einstein summation convention, I appreciated the fact that although the author mentioned it (albeit without using the name “Einstein”), he deliberately chose not to use it in this chapter). Some ideas of non-Euclidean geometry (the Poincare half-plane) are discussed here, and the idea of a Riemann surface is introduced.

The remaining chapter addresses the global theory of surfaces, the highlight of which is perhaps the statement and proof of the Gauss-Bonnet theorem. The chapter ends with a brief foray into algebraic topology, via the Euler-Poincaré characteristic.

Two multi-section appendices, one on basic topology and the other on differential equations, round out the text. The first of these appendices starts with the basic constructs of topology in Euclidean spaces (open and closed sets, connectedness, compactness, etc.) and then generalizes them to arbitrary topological spaces. Full proofs are generally given, even of some deep results like the Heine-Borel theorem. The second appendix, on differential equations, is much shorter and generally devoid of proofs; it simply states some results on both ordinary and partial differential equations that were used in the text.

American undergraduate mathematics majors don’t generally get to take courses in this material much anymore, so, even though this book comes equipped with exercises, it probably won’t find much use as an undergraduate text; Springer doesn’t seem to even be marketing it as one, since the webpage for the book does not list it under “textbooks” and lists the level as “graduate”. This is probably appropriate — Professor Borceux teaches in Europe (in Belgium) and this book was undoubtedly written for European undergraduates. While it is clearly written, it is fairly dense and requires some degree of mathematical maturity.

The fact that this may not be used much as a textbook, however, does not detract from its quality. This trilogy is, in fact, a remarkable accomplishment; Borceux obviously cares a lot about geometry, and these books are a real labor of love. I’m glad that they are on my shelves; if you have any interest in geometry, they should be on yours as well.

One final comment, unrelated to the quality of this particular book, concerns Springer’s production practices. I recently had to return a Springer book that I purchased on amazon because my copy had about 50 blank pages in it, including one streak of 17 consecutive blank pages; a colleague I emailed reported similar printing problems with other books. In this volume, none of the pages were blank, but one did fall out of the book as I opened it for the first time.

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