**Graduate Studies in Mathematics**

Volume: 69;
2009;
376 pp;
Hardcover

MSC: Primary 53;

Print ISBN: 978-0-8218-4763-3

Product Code: GSM/69.R

List Price: $73.00

Individual Member Price: $58.40

**Electronic ISBN: 978-1-4704-1151-0
Product Code: GSM/69.R.E**

List Price: $73.00

Individual Member Price: $58.40

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# Curves and Surfaces: Second Edition

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*Sebastián Montiel; Antonio Ros*

This introductory textbook puts forth a clear and focused point of view on
the differential geometry of curves and surfaces. Following the modern
point of view on differential geometry, the book emphasizes the global
aspects of the subject. The excellent collection of examples and exercises
(with hints) will help students in learning the material. Advanced
undergraduates and graduate students will find this a nice entry point to
differential geometry.

In order to study the global properties of curves and surfaces, it is
necessary to have more sophisticated tools than are usually found in
textbooks on the topic. In particular, students must have a firm grasp on
certain topological theories. Indeed, this monograph treats the
Gauss–Bonnet theorem and discusses the Euler characteristic. The authors
also cover Alexandrov's theorem on embedded compact surfaces in
\(\mathbb{R}^3\) with constant mean curvature. The last chapter
addresses the global geometry of curves, including periodic space curves and
the four-vertices theorem for plane curves that are not necessarily convex.

Besides being an introduction to the lively subject of curves and
surfaces, this book can also be used as an entry to a wider study of
differential geometry. It is suitable as the text for a first-year
graduate course or an advanced undergraduate course.

This book is published in cooperation with Real Sociedad Matemática Española (RSME)

#### Table of Contents

# Table of Contents

## Curves and Surfaces: Second Edition

- Cover Cover11 free
- Title page iii5 free
- Contents vii9 free
- Preface to the second edition xi13 free
- Preface to the English edition xiii15 free
- Preface xv17 free
- Plane and space curves 119 free
- Surfaces in Euclidean space 3149
- The second fundamental form 6785
- Separation and orientability 107125
- Integration on surfaces 135153
- Global extrinsic geometry 171189
- Intrinsic geometry of surfaces 203221
- The Gauss–Bonnet theorem 275293
- Global geometry of curves 309327
- Bibliography 371389
- Index 373391 free
- Back Cover Back Cover1395

- Cover Cover1396
- Title page iii400
- Contents vii404
- Preface to the second edition xi408
- Preface to the English edition xiii410
- Preface xv412
- Plane and space curves 1414
- Surfaces in Euclidean space 31444
- The second fundamental form 67480
- Separation and orientability 107520
- Integration on surfaces 135548
- Global extrinsic geometry 171584
- Intrinsic geometry of surfaces 203616
- The Gauss–Bonnet theorem 275688
- Global geometry of curves 309722
- Bibliography 371784
- Index 373786
- Back Cover Back Cover1790

#### Readership

Undergraduate students, graduate students, and research mathematicians interested in the geometry of curves and surfaces.

#### Reviews

This book is a nice introduction to differential geometry with contemporary emphasis on aspects of a global nature.

-- Mathematical Reviews

With its readable style and the completeness of its exposition, this would be a very good candidate for an introductory graduate course in differential geometry or for self-study.

-- MAA Reviews