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