Preface

This book is based on notes I wrote when teaching an undergraduate

seminar on surfaces at Brown University in 2005. Each week I wrote

up notes on a different topic. Basically, I told the students about

many of the great things I have learned about surfaces over the years.

I tried to do things in as direct a fashion as possible, favoring concrete

results over a buildup of theory. Originally, I had written 14 chapters,

but later I added 9 more chapters so as to make a more substantial

book.

Each chapter has its own set of exercises. The exercises are em-

bedded within the text. Most of the exercises are fairly routine, and

advance the arguments being developed, but I tried to put a few

challenging problems in each batch. If you are willing to accept some

results on faith, it should be possible for you to understand the mate-

rial without working the exercises. However, you will get much more

out of the book if you do the exercises.

The central object in the book is a surface. I discuss surfaces

from many points of view: as metric spaces, triangulated surfaces,

hyperbolic surfaces, and so on. The book has many classical results

about surfaces, both geometric and topological, and it also has some

extraneous stuff that I included because I like it. For instance, the

book contains proofs of the Pythagorean Theorem, Pick’s Theorem,

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