Mostly Surfaces

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.

Cover 1 Title page 2 Contents 4 Preface 12 Book overview 16 Part 1. Surfaces and topology 34 Definition of a surface 36 The gluing construction 46 The fundamental group 58 Examples of fundamental groups 68 Covering spaces and the deck group 80 Existence of universal covers 94 Part 2. Surfaces and geometry 100 Euclidean geometry 102 Spherical geometry 118 Hyperbolic geometry 130 Riemann metrics on surfaces 148 Hyperbolic surfaces 158 Part 3. Surfaces and complex analysis 176 A primer on complex analysis 178 Disk and plane rigidity 192 The Schwarz-Christoffel transformation 198 Riemann surfaces and uniformization 210 Part 4. Flat cone surfaces 220 Flat cone surfaces 222 Translation surfaces and the Veech group 236 Part 5. The totality of surfaces 252 Continued fractions 254 Teichmüller space and moduli space 266 Topology of Teichmüller space 278 Part 6. Dessert 288 The Banach Tarski theorem 290 Dehn’s dissection theorem 302 The Cauchy rigidity theorem 310 Bibliography 324 Index 326 Back Cover 330

Richard Evan Schwartz: Brown University, Providence, RI

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