Lectures on Surfaces: (Almost) Everything You Wanted to Know about Them does a masterful job of introducing the study of surfaces to advanced undergraduates. Although there are many attractive volumes in the AMS Student Mathematical Library series, this is the first one I’ve seen that would really have captured my interest as an undergraduate.
One of the reasons why this text works so well (I think) is that the authors are not experts in the area. They take extra care to elaborate or explain things that an expert would not, and then anticipate those places where a newcomer might get stuck. Nonetheless, their scope is ambitious and ranges all the way from triangulation and classification of surfaces to Riemann surfaces, Riemannian geometry on surfaces, and the Gauss-Bonnet theorem. The Euler characteristic in its many guises is ubiquitous.
The book is divided into five chapters consisting of thirty-six lectures. Since this work developed from the MASS (Mathematics Advanced Study Semesters) program at Penn State, the lecture subdivision is a natural consequence of how material was divided for presentation in the classroom. The authors assume that students’ background includes the usual calculus sequence, basic linear algebra, rudimentary differential equations, and a bit of real analysis. As important as the prerequisites is an appetite for learning new mathematics at a pretty brisk pace.
The authors begin with basic examples of surfaces and describe various ways of representing surfaces: by an equation, by planar model and quotient space, by local coordinates, or parametrically. The second chapter focuses on the combinatorial structure and topological characterization of surfaces. This includes classification of compact surfaces, with a proof for the orientable case. The authors introduce triangulations and the Euler characteristic of a triangulation; then they go on to define homology groups and Betti numbers and so provide a second interpretation of the Euler characteristic. The third and fourth chapters take up differentiable structures and vector fields on surfaces, Riemann surfaces, and then Riemann metrics. This brings us to geodesics, curvature, the hyperbolic plane, and the Gauss-Bonnet theorem. The final chapter comes back to topology and smooth structures for a discussion of degree and index of vector fields.
It is amazing how much mathematics is naturally associated with the study of surfaces, and how the pieces fit together so remarkably. The authors succeed in pulling in many topics while keeping their story coherent and compelling. This book would work well as the text for a capstone course or independent reading. However, there are relatively few exercises, so an instructor would probably need to develop supplementary problem sets for classroom use.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.