Geometry of Surfaces

Rating:
Summary: Interesting advanced undergraduate course
Review: Stillwell contends (in his preface) that the geometry of surfaces of constant curvature is an ideal topic for such a course, and he gives three convincing reasons for that, the most important one being "maximal connectivity with the rest of mathematics," which he elucidates. I applaud this.

He then demurs that such a deep and broad topic cannot be covered completely by a book of his modest size. He does include, at the end of each chapter, informal discussions of further results and references to the literature - these are very valuable.

The teacher of the teacher of Stillwell's teacher was Felix Klein, and Stillwell approaches his subject in the spirit of Klein. His first chapter describes in detail the group of isometries of the Euclidean plane E. Then his second chapter gives the Hopf-Killing classification of complete, connected Euclidean surfaces as quotient spaces of E by certain groups of isometries of E, and up to isometry there are exactly five such (cylinder, twisted cylinder, torus, Klein bottle and E itself). The proof introduces the student to the important subject of covering spaces.

Stillwell's writing style is pleasantly informal but can be careless. The main subject of the book is surfaces, but he never defines "surface!" He does define the compound "Euclidean surface," but his definition is inadequate: he doesn't require that his distance function only take on positive real values for distinct points, and he doesn't specify the conditions that it be a metric (e.g., triangle inequality). Evidently a Euclidean surface is a metric space that is locally isometric to E.

The next two chapters are very good introductions to two-dimensional spherical, elliptic and hyperbolic geometries, again with a description of their isometries. The hyperbolic plane is introduced by first showing nicely that the pseudosphere has Gaussian curvature -1, and then transferring a suitable coordinate system and infinitesimal distance function on the pseudosphere over to the upper half-plane H.

Stillwell asserts without proof that Gaussian curvature is well-defined (for "surfaces" in Euclidean three-space); he gives no reference for that result. He does not mention Gauss' Theorema Egregrium either. In fact he pretty much skirts differential geometry altogether in this book.

The meat of the book is chapter 5 on hyperbolic surfaces (metric spaces which are locally isometric to H). He states without proof Rado's theorem that any compact surface is homeomorphic to the identification space of a polygon (he doesn't explain that "surface" in this theorem means two-dimensional topological manifold). He applies this result to show that such surfaces can be "realized geometrically". He doesn't define that either, but from his argument we glean that such topological surfaces can underly a structure of either Euclidean, hyperbolic or spherical surface (locally isometric to the sphere S).

Chapter 6 begins with the classification of compact topological surfaces and their fundamental groups. For a "geometric surface" X, which now means a quotient of either E, H or S by a discontinuous fixed-point-free group G of isometries, he proves that G is isomorphic to the fundamental group of X. He is able to define a "geodesic path" on X without using differential geometry, but warns of difficulties with "geodesic monogons." He proves that on a compact orientable surface of genus > 1, each non-trivial free homotopy class has a unique geodesic representative.

The final two chapters are a nice treatment of tessellations.

In sum, this book is a very good introduction for advanced undergraduates to the portion of surface geometry that interests Stillwell. It is an attractive mixture of topology, algebra and a smidgen of analysis.