**Graduate Studies in Mathematics**

Volume: 61;
2003;
378 pp;
Hardcover

MSC: Primary 53;
Secondary 58; 35; 14

Print ISBN: 978-0-8218-3375-9

Product Code: GSM/61

List Price: $69.00

Individual Member Price: $55.20

**Electronic ISBN: 978-1-4704-1801-4
Product Code: GSM/61.E**

List Price: $69.00

Individual Member Price: $55.20

#### You may also like

#### Supplemental Materials

# Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems

Share this page
*Thomas A. Ivey; J. M. Landsberg*

This book is an introduction to Cartan's approach to differential geometry. Two central methods in Cartan's geometry are the theory of exterior differential systems and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems.

It begins with the classical geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally with motivating examples leading to definitions, theorems, and proofs.

Once the basics of the methods are established, the authors develop applications and advanced topics. One notable application is to complex algebraic geometry, where they expand and update important results from projective differential geometry.

The book features an introduction to \(G\)-structures and a treatment of the theory of connections. The Cartan machinery is also applied to obtain explicit solutions of PDEs via Darboux's method, the method of characteristics, and Cartan's method of equivalence.

This text is suitable for a one-year graduate course in differential geometry, and parts of it can be used for a one-semester course. It has numerous exercises and examples throughout. It will also be useful to experts in areas such as PDEs and algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields.

#### Table of Contents

# Table of Contents

## Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface ix10 free
- Moving frames and exterior differential systems 116 free
- Euclidean geometry and Riemannian geometry 3550
- Projective geometry 7186
- Cartan-Kähler I: Linear algebra and constant-coefficient homogeneous systems 143158
- Cartan-Kähler II: The Cartan algorithm for linear Pfaffian systems 163178
- Applications to PDE 203218
- Cartan-Kähler III: The general case 243258
- Geometric structures and connections 267282
- Linear algebra and representation theory 311326
- Differential forms 335350
- Complex structures and complex manifolds 343358
- Initial value problems 349364
- Hints and answers to selected exercises 355370
- Bibliography 363378
- Index 371386 free
- Back Cover Back Cover1394

#### Readership

Graduate students and research mathematicians interested in differential geometry and exterior differential systems.