Curved Spaces
From Classical Geometries to Elementary Differential Geometry

This 2007 textbook uses examples, exercises, diagrams, and unambiguous proof, to help students make the link between classical and differential geometries.

P. M. H. Wilson (Author)

9780521713900, Cambridge University Press

Paperback, published 13 December 2007

198 pages, 79 b/w illus. 105 exercises
24.4 x 17 x 1.1 cm, 0.33 kg

'The book is written in a nice and precise style and explicit computations and proofs make the book easy to understand. A detailed and explicit discussion of the main examples of classical geometries contributes well to a better understanding of later generalisations. A list of examples at the end of each chapter helps as well. It is a very good addition to the literature on the topic and can be very useful for teachers preparing their courses as well as for students.' EMS Newsletter

This self-contained 2007 textbook presents an exposition of the well-known classical two-dimensional geometries, such as Euclidean, spherical, hyperbolic, and the locally Euclidean torus, and introduces the basic concepts of Euler numbers for topological triangulations, and Riemannian metrics. The careful discussion of these classical examples provides students with an introduction to the more general theory of curved spaces developed later in the book, as represented by embedded surfaces in Euclidean 3-space, and their generalization to abstract surfaces equipped with Riemannian metrics. Themes running throughout include those of geodesic curves, polygonal approximations to triangulations, Gaussian curvature, and the link to topology provided by the Gauss-Bonnet theorem. Numerous diagrams help bring the key points to life and helpful examples and exercises are included to aid understanding. Throughout the emphasis is placed on explicit proofs, making this text ideal for any student with a basic background in analysis and algebra.

Preface
1. Euclidean geometry
2. Spherical geometry
3. Triangulations and Euler numbers
4. Riemannian metrics
5. Hyperbolic geometry
6. Smooth embedded surfaces
7. Geodesics
8. Abstract surfaces and Gauss-Bonnet.

Subject Areas: Differential & Riemannian geometry [PBMP], Calculus & mathematical analysis [PBK]