Mostly Surfaces (Student Mathematical Library, 60)

This book presents a number of topics related to surfaces, such as Euclidean, spherical and hyperbolic geometry, the fundamental group, universal covering surfaces, Riemannian manifolds, the Gauss-Bonnet Theorem, and the Riemann mapping theorem. The main idea is to get to some interesting mathematics without too much formality. The book also includes some material only tangentially related to surfaces, such as the Cauchy Rigidity Theorem, the Dehn Dissection Theorem, and the Banach–Tarski Theorem. The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis. This book is based on notes that Schwartz made when he taught an undergraduate seminar on surfaces at Brown University in 2005. Students should have completed a semester each in real analysis, abstract algebra, and complex analysis, though he embeds a kind of mini-course on complex analysis in this text. He discusses surfaces from many perspectives, such as metric spaces, triangulated surfaces, and hyperbolic surfaces as well as some matter only tangential to surfaces. Annotation 2011 Book News, Inc., Portland, OR (booknews.com) The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis. --from publisher description.