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.

Richard Evan Schwartz : Brown University, Providence, RI

Chapter 1. Book overview
Part 1. Surfaces and topology

Chapter 2. Definition of a surface

Chapter 3. The gluing construction

Chapter 4. The fundamental group

Chapter 5. Examples of fundamental groups

Chapter 6. Covering spaces and the deck group

Chapter 7. Existence of universal covers
Part 2. Surfaces and geometry

Chapter 8. Euclidean geometry

Chapter 9. Spherical geometry

Chapter 10. Hyperbolic geometry

Chapter 11. Riemannian metrics on surfaces

Chapter 12. Hyperbolic surfaces
Part 3. Surfaces and complex analysis

Chapter 13. A primer on complex analysis
Chapter 14. Disk and plane rigidity

Chapter 15. The Schwarz-Christoffel transformation

Chapter 16. Riemann surfaces and uniformization
Part 4. Flat cone surfaces
Chapter 17. Flat cone surfaces

Chapter 18. Translation surfaces and the Veech group
Part 5. The totality of surfaces

Chapter 19. Continued fractions

Chapter 20. Teichmüller space and moduli space

Chapter 21. Topology of Teichmüller space
Part 6. Dessert

Chapter 22. The Banach–Tarski theorem

Chapter 23. Dehn’s dissection theorem

Chapter 24. The Cauchy rigidity theorem