Curved Spaces: From Classical Geometries to Elementary Differential Geometry

by P.M.H. Wilson

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.

Genres: Mathematics, Textbooks, Geometry

198 pages, Paperback

First published January 14, 2007

Reviews

[Medhat ullah · Rating 5 out of 5]

Exposition of well-known classical dimensional geometries, such as Euclidean, spherical, hyperbolic & Euler numbers for topological triangulations, Riemannian metrics, Euclidean 3-spaces, abstract Surfaces, geodesics curves, polygonal approximation, gaussian curvature & bit differential geometry