Geometry of Surfaces (Universitext) by John Stillwell (PDF)

Description

The geometry of surfaces is an ideal starting point for learning geometry, for, among other reasons, the theory of surfaces of constant curvature has maximal connectivity with the rest of mathematics. This text provides the student with the knowledge of a geometry of greater scope than the classical geometry taught today, which is no longer an adequate basis for mathematics or physics, both of which are becoming increasingly geometric. It includes exercises and informal discussions.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book covers an interesting selection of topics (as Prof. Goodman describes in his excellent review), but I wished for a more engaging presentation. The exposition is often rather concise, frequently no more than a page or two followed by exercises. Since the results of the exercises are often relied on in the sequel, the reader acutely feels the absence of solutions, hints and closed-form (“show that…” type) problems. The book is moderately illustrated, but not quite adequately for an undergraduate textbook. Its description of isometries in hyperbolic geometry relies heavily on words and algebra, even though these transformations are very difficult for a beginner to visualize. Ditto for even the humble Möbius transformation, for which Wikipedia will give you a better gut-level understanding. The notational choices, especially using ‘r’ for both rotations and reflections (albeit in the latter case with a bar over it), are sometimes unfortunate.The author claims that basic linear algebra and calculus, along with dollops of group theory and point-set topology (open, closed and compact sets) are sufficient background. But familiarity with a bit more topology (e.g., homotopy and a basic understanding the idea of genus) will help, since the book’s treatment of the topological material is lightning-fast. And having a course in complex variables under one’s belt will make it much easier to appreciate what’s so beautiful and unifying about the material presented here.To calibrate: I’m not a mathematician or in school; this was purely hobby reading. I had hoped that this would be a relaxing read during a snowy year-end holiday, but I found myself getting annoyed with the author’s not-quite rigorous, not-quite user-friendly approach. The book is already 20 years old, so maybe it’s time for a revision — especially since math graphics software is now widely available, and Springer has begun printing in color. I hope a new edition could be produced with more informative illustrations and at least a partial set of solutions or hints. For now: a 1 star deduction for the total absence of aids to exercises, and another minus .5 for the stylistic stuff.

⭐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.

⭐This is a great book as an introduction to modern geometry for college students without much background in advanced math (like differential geometry, manifolds, topology, etc). I used it as textbook when teaching the course modern geometry and felt really happy about the content (though some proofs could be simplified).

⭐My math background is two semesters of real analysis, one of complex analysis, and some group theory. I found this book to be pretty accessible; the concepts are all clearly explained and usually accompanied by informative figures. I had to look up a number of the terms and wasn’t familiar with all of the topology, but without much work I at least gleaned the underlying concepts, which generally blew my mind. I do a lot of independent reading on math and have found that this book familiarized me with many important concepts like group actions and homotopy.The style is pretty informal. Lots of pictures, lots of explaining, which is important to me. Stillwell structures each chapter nicely with a little preliminary discussion of what “problems” he’s about to solve, goes through proofs and explanations, then ties up the chapter with a discussion relating that chapter to previous/future material and often to other branches of math. This was my first taste of geometry so I don’t know any standard texts to compare it to. My impression was that this book is good for getting the ideas under your belt, but maybe doesn’t go into great depth. I was also pretty impressed by how cheap it was in paperback.

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