Curves and Surfaces (Graduate Studies in Mathematics) 2nd Edition by Sebastian Montiel and Antonio Ros (PDF)

Description

This introductory textbook puts forth a clear and focused point of view on the differential geometry of curves and surfaces. Following the modern point of view on differential geometry, the book emphasizes the global aspects of the subject. The excellent collection of examples and exercises (with hints) will help students in learning the material. Advanced undergraduates and graduate students will find this a nice entry point to differential geometry. In order to study the global properties of curves and surfaces, it is necessary to have more sophisticated tools than are usually found in textbooks on the topic. In particular, students must have a firm grasp on certain topological theories. Indeed, this monograph treats the Gauss-Bonnet theorem and discusses the Euler characteristic. The authors also cover Alexandrov’s theorem on embedded compact surfaces in $mathbb{R}^3$ with constant mean curvature. The last chapter addresses the global geometry of curves, including periodic space curves and the four-vertices theorem for plane curves that are not necessarily convex. Besides being an introduction to the lively subject of curves and surfaces, this book can also be used as an entry to a wider study of differential geometry. It is suitable as the text for a first-year graduate course or an advanced undergraduate course.

User’s Reviews

Editorial Reviews: Review With its readable style and the completeness of its exposition, this would be a very good candidate for an introductory graduate course in differential geometry or for self-study. –MAA ReviewsThis book is a nice introduction to differential geometry with contemporary emphasis on aspects of a global nature. –Mathematical Reviews

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

⭐The only exception to completeness of the book is that knowledge of Lebesgue integration is assumed. The text is a careful and explicit treatment of its topic. Derivations of first variations of enclosed volume, surface area and length of curves in surfaces are included in full and honest detail. Everything is done in Euclidean space although quantities are classified as intrinsic or extrinsic. Gauss-Bonnet and other major theorems are proven from first principles. The authors did not shy away from sophisticated topics. Many details my major professor assumed I knew are given here. Great exercises with hints. One of my favorite little books.

⭐We used this book for our first semester in differential geometry. To be a little honest, I felt a tad bit betrayed. This book just seemed to me to be glorified calculus III. I could teach the material from this book, easy peasy.The book is good, though I think the authors will have trouble finding an appropriate audience. I think that it would be good for the advanced junior or senior undergraduate. I think a graduate student needs something more…. Whatever that will be will depend on the needs of the student.

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