Geometry of Surfaces

"Geometry of Surfaces explores the interplay between geometry and topology in the simplest nontrivial case : the surfaces of constant curvature. As such, it provides a concise introduction to modern geometry for a wide audience. Requiring only a little prior knowledge of undergraduate mathematics, the book begins by discussing the three simplest surfaces : the Euclidean plane (zero curvature), the sphere (positive curvature), and the hyperbolic plane (negative curvature). Using the efficient machinery of isometry grouops, the author extends the discussion to all surfaces of constant curvature, which are typically obtained from the simplest ones by suitable isometries. The book then turns to the classification of the finitely many Euclidean and spherical surfaces and to a study of some remarkable hyperbolic surfaces. The general problem of classification is then considered from a topological and group-theoretic viewpoint. Because the theory of surfaces of constant curvature is intimately connected with the rest of modern mathematics, this book is an ideal starting point for students learning geometry, providing the simplest possible introduction to curvature, group actions, and covering spaces. The concepts developed here are, historically, the source of many concepts of complex analysis, differential geometry, topology, and combinatorial group theory, as well as such hot topics as fractal geometry and string theory. The prerequisites are modest, including only a little linear algebra, calculus, basic group theory, and basic topology. The formal coverage is extended by exercises and informal discussions throughout the text."--taken from back cover.