Softcover ISBN:  9780821853689 
Product Code:  STML/60 
List Price:  $52.00 
Individual Price:  $41.60 
Electronic ISBN:  9781470412234 
Product Code:  STML/60.E 
List Price:  $49.00 
Individual Price:  $39.20 

Book DetailsStudent Mathematical LibraryVolume: 60; 2011; 314 ppMSC: Primary 14; 30; 32; 37; 53; 51;
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 GaussBonnet 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.ReadershipUndergraduate students interested in geometry and topology of surfaces.

Table of Contents

Chapters

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 SchwarzChristoffel 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


Additional Material

Reviews

The book contains a lot of interesting basic and more advanced material which is presented in a nice, intuitive yet rigorous way, and, as such, is perfectly suited as an accompanying text or additional reading for a first course on topology or as a basis for a student seminar.
Mathematical Reviews 
This is a novel, eclectic, and ambitious collection of geometric and topological topics developed as they relate to surfaces ... a terrific volume. Highly recommended.
CHOICE 
...a delightful reading. Schwartz gives a beautiful, careful exposition of some of the most elegant ideas, theorems, and proofs in the theory of surfaces. It's an ideal book for casual reading in spare mathematical moments.
MAA Reviews 
This highly readable book is an excellent introduction to the theory of surfaces, covering a wide variety of topics with references for further reading. Each chapter contains numerous exercises on the material to get the reader thinking about the subjects covered. There are also many diagrams to aid the reader in understanding the material.
Alastair Fletcher, Zentralblatt MATH


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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 GaussBonnet 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.
Undergraduate students interested in geometry and topology of surfaces.

Chapters

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 SchwarzChristoffel 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

The book contains a lot of interesting basic and more advanced material which is presented in a nice, intuitive yet rigorous way, and, as such, is perfectly suited as an accompanying text or additional reading for a first course on topology or as a basis for a student seminar.
Mathematical Reviews 
This is a novel, eclectic, and ambitious collection of geometric and topological topics developed as they relate to surfaces ... a terrific volume. Highly recommended.
CHOICE 
...a delightful reading. Schwartz gives a beautiful, careful exposition of some of the most elegant ideas, theorems, and proofs in the theory of surfaces. It's an ideal book for casual reading in spare mathematical moments.
MAA Reviews 
This highly readable book is an excellent introduction to the theory of surfaces, covering a wide variety of topics with references for further reading. Each chapter contains numerous exercises on the material to get the reader thinking about the subjects covered. There are also many diagrams to aid the reader in understanding the material.
Alastair Fletcher, Zentralblatt MATH