Lectures on Surfaces (Student Mathematical Library) by Anatole Katok and Vaughn Climenhaga (PDF)

Description

Surfaces are among the most common and easily visualized mathematical objects, and their study brings into focus fundamental ideas, concepts, and methods from geometry, topology, complex analysis, Morse theory, and group theory. At the same time, many of those notions appear in a technically simpler and more graphic form than in their general “natural” settings. The first, primarily expository, chapter introduces many of the principal actors–the round sphere, flat torus, Mobius strip, Klein bottle, elliptic plane, etc.–as well as various methods of describing surfaces, beginning with the traditional representation by equations in three-dimensional space, proceeding to parametric representation, and also introducing the less intuitive, but central for our purposes, representation as factor spaces. It concludes with a preliminary discussion of the metric geometry of surfaces, and the associated isometry groups. Subsequent chapters introduce fundamental mathematical structures–topological, combinatorial (piecewise linear), smooth, Riemannian (metric), and complex–in the specific context of surfaces. The focal point of the book is the Euler characteristic, which appears in many different guises and ties together concepts from combinatorics, algebraic topology, Morse theory, ordinary differential equations, and Riemannian geometry. The repeated appearance of the Euler characteristic provides both a unifying theme and a powerful illustration of the notion of an invariant in all those theories. The assumed background is the standard calculus sequence, some linear algebra, and rudiments of ODE and real analysis. All notions are introduced and discussed, and virtually all results proved, based on this background. This book is a result of the MASS course in geometry in the fall semester of 2007.

User’s Reviews

Editorial Reviews: Review (This book) does a masterful job of introducing the study of surfaces to advanced undergraduates. … 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. —– MAA ReviewsThis book will be a welcome addition to college and university libraries and an excellent source for supplementary reading. —- Mathematical Reviews

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

⭐This book would be helpful if you want to put in order surface theory in your mind. But there is some ambiguity about the audience of this book. If you are interested in surface theory as an undergraduate student, this book would be hard. If you are a post-doc, many parts of this book would be already familiar to you. But it was good for me though I am a post-doc.The whole title of this book is Lectures on Surfaces: (Almost) Everything You Wanted To Know About Them. But it seems to be that the subtitle is a little bit exaggerated. For example, only a little attention is given to surface with boundary or with puncture, and there is no argument about surface automorphisms and surfaces embedded or immersed in higher dimensional manifolds. And there are only basics about Teichmuller space. But still, I believe that this is an appropriate title.The best thing for me in this book was an understanding of the meaning of the following theorem: given a triangulation of a surface, there exists a smooth atlas on the surface, and vice versa (page 112). If you are a math-graduate student, then you may have heard of the theorem and take it as a fact. But do you exactly understand what the theorem means? Have you ever seen proofs of it? I do not say a proof is important (of course, it is important), but the precise-understanding of its content is important. If you are going to read this book, please follow carefully how the authors dealt with it. They don’t give a full proof of it. But it suffices to understand the meaning of the theorem.I’d like to give you, a future reader, a piece of advice about how to read this book efficiently. First, you would better read this book within two or three weeks. In my opinion, this is not the kind of the textbooks in math curriculum. Reading such books takes usually several months with thorough analysis and lots of exercises. Many exercises are shown in this book, but never essential parts of it. For example, the exercise 3.4 on page 116 is not used afterwards. If you don’t feel like solving it, just skip it. To me, the most boring part was chapter 2. In reading such chapters, just read lightly and move forward. If it happens that you need to understand what you skipped, then you can go back to that point anytime. Moreover, the chapters are somewhat independent.Secondly, don’t be too serious about this book. The most required prerequisite to read this book is covering space theory, although it is explained a little in this book. If you haven’t studied covering space theory, then this book can be hard to you. Also in that case, just move forward and don’t be too serious about your ability of learning mathematics.Now I want to tell you the good and bad points of this book according to page order.1. (Page 120) There is an argument how to remove a singularity when we make a smooth surface from identifying pairs of sides of a polygon. To me, it was insufficient. I’d like to ask them (the two authors) to explain more logically and precisely or show us a full proof of it.2. (Page 126) The proof of proposition 3.7 is not understandable, although it is an important theorem of this book. I hope that the authors add more explanations on the proof.3. (Page 130) If you heard of Teichmuller space but feels it is too abstract, then I recommend that you read this part of the book. It is really reasonably well-written from the very basic. Of course, this book deals with a lot of topics of surfaces, so it doesn’t go deeply about Teichmuller space.4. (Page 194) It deals with the linear fractional transformations. When I first read this part, I tried to skip it because it seems to overlap with the articles and books that I have read about complex analysis and hyperbolic geometry. But while I read it, I got the impression that the authors really understand plane hyperbolic geometry well, and the understanding pervades me. I want to add one more comment. In this part, when the author says “a linear fractional transformation”, sometimes it is not obvious in sentences in its literal form whether he means a linear fractional transformation with real coefficients or complex coefficients. Readers should be careful in detecting their distinction because it’s crucial to exactly understand plane hyperbolic geometry.5. (Chapter 5) In my opinion, many parts of chapter 5 of this book is not satisfactory. Its content is OK, but the proofs seems to be a little sloppy. Above all, the highlight of chapter 5 should be the Jordan curve theorem, but the proof does not seem reasonable. The proof of the Smooth Jordan curve theorem was amazingly beautiful, but not so with the Jordan curve theorem. After the disappointment, I didn’t feel like goning to Poincare-Hopf index formula.As a whole, it was an exciting experience to read this book.

⭐This book is mainly established on the lecture notes of the course “Surfaces” in the MASS program 2007 at Penn State. As a participant in the course, I hereby give my strongest rate to memorise that unforgettable semester.I can not think, any other course, can be comparable to this one which gives an innovative, comprehensiveintroduction to the mathematical theory of surfaces in such a genius way. Thanks for the lecturer Professor Katokand also our best TA Vaughn.

⭐It is easy to lose one’s way when faced with the myriad of mathematical techniques that one could apply to the workings of the brain. This extensively illustrated book gently guides the reader through the different geometric approaches, many of which have already been applied to neural mechanisms. The authors concentrate on surfaces which enables them to give relatively simple explanations and proofs so that the reader gains a clear impression of what would be involved in taking it further with a particular technique. More than that, the authors have unified the approaches by using each of them to reveal an alternative formulation of the Euler characteristic of the surface. In its simplest form the Euler characteristic relates the number of faces, edges and vertices of a polyhedron. But that is just number one of the seven alternative formulations the authors arrive at by the end of the book. There are some of the out-of-sequence parts and the index is minimalist so that you have to read it more than once to become familiar with where all the definitions are, but maybe that is no bad thing.

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