**Student Mathematical Library**

Volume: 46;
2008;
286 pp;
Softcover

MSC: Primary 51; 53; 57;

Print ISBN: 978-0-8218-4679-7

Product Code: STML/46

List Price: $50.00

Individual Price: $40.00

**Electronic ISBN: 978-1-4704-1817-5
Product Code: STML/46.E**

List Price: $50.00

Individual Price: $40.00

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#### Supplemental Materials

# Lectures on Surfaces: (Almost) Everything You Wanted to Know about Them

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*Anatole Katok; Vaughn Climenhaga*

Surfaces are among the most common and easily visualized mathematical
objects, and their study brings into focus fundamental ideas, concepts, and
methods from geometry, topology, complex analysis, Morse theory, and
group theory. At the same time, many of those notions appear in a
technically simpler and more graphic form than in their general
“natural” settings.

The first, primarily expository, chapter introduces many of the
principal actors—the round sphere, flat torus, Möbius strip, Klein
bottle, elliptic plane, etc.—as well as various methods of
describing surfaces, beginning with the traditional representation by
equations in three-dimensional space, proceeding to parametric
representation, and also introducing the less intuitive, but central
for our purposes, representation as factor spaces. It concludes with
a preliminary discussion of the metric geometry of surfaces, and the
associated isometry groups. Subsequent chapters introduce fundamental
mathematical structures—topological, combinatorial
(piecewise linear), smooth, Riemannian (metric), and complex—in the
specific context of surfaces.

The focal point of the book is the Euler characteristic, which appears in
many different guises and ties together concepts from combinatorics,
algebraic topology, Morse theory, ordinary differential equations, and
Riemannian geometry. The repeated appearance of the Euler characteristic
provides both a unifying theme and a powerful illustration of the notion
of an invariant in all those theories.

The assumed background is the standard calculus sequence, some linear
algebra, and rudiments of ODE and real analysis. All notions are
introduced and discussed, and virtually all results proved, based on this
background.

This book is a result of the MASS course in geometry in the fall semester
of 2007.

This book is published in cooperation with Mathematics Advanced Study Semesters

#### Table of Contents

# Table of Contents

## Lectures on Surfaces: (Almost) Everything You Wanted to Know about Them

- Cover Cover11 free
- Title page iii5 free
- Contents v7 free
- Foreword: MASS and REU at Penn State University xi13 free
- Preface xiii15 free
- Various ways of representing surfaces and basic examples 119 free
- Combinatorial structure and topological classification of surfaces 4967
- Differentiable structure on surfaces: Real and complex 103121
- Riemannian metrics and geometry of surfaces 159177
- Topology and smooth structure revisited 243261
- Suggested reading 271289
- Hints 275293
- Index 283301 free
- Back Cover Back Cover1307

#### Readership

Undergraduate and graduate students interested in broadening their view of geometry and topology.

#### Reviews

This book will be a welcome addition to college and university libraries and an excellent source for supplementary reading.

-- Mathematical Reviews

(This book) does a masterful job of introducing the study of surfaces to advanced undergraduates. ... The authors succeed in pulling in many topics while keeping their story coherent and compelling. This book would work well as the text for a capstone course or independent reading.

-- MAA Reviews