Curved Spaces

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.

• A concrete approach to the theory, with emphasis on self-contained explicit proofs; uses the classical geometries to motivate the basic ideas of elementary differential geometry
• Provides a link between basic undergraduate courses on Analysis and Algebra, and more advanced theoretical courses in geometry
• Rigorous treatment of the classical geometries, via analytical ideas, with exercises at the end of each chapter, reinforcing the material in the text
• A novel approach to defining curvature on abstract surfaces, and to proving the topological invariance of the Euler number
• Coverage of a wide range of topics, starting with very elementary material and concluding with rather more advanced mathematical ideas
• Certain geometrical themes, such as geodesics, curvature, and the Gauss-Bonnet theorem, running throughout the book, provide a unifying philosophy