Graduate Studies in Mathematics
Volume: 61;
2003;
378 pp;
MSC: Primary 53;
Secondary 58; 35; 14
Electronic ISBN: 978-1-4704-1801-4
Product Code: GSM/61.E
List Price: $69.00
AMS Member Price: $55.20
MAA Member Price: $62.10
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Supplemental Materials
Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems
Share this pageThomas 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