Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems
About this Title
Thomas A. Ivey, College of Charleston, Charleston, SC and J. M. Landsberg, Georgia Institute of Technology, Atlanta, GA
Publication: Graduate Studies in Mathematics
Publication Year: 2003; Volume 61
ISBNs: 978-0-8218-3375-9 (print); 978-1-4704-1801-4 (online)
MathSciNet review: MR2003610
MSC: Primary 53-01; Secondary 58A15
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. The book presents thorough and modern treatments of both subjects, including their applications to classic and contemporary problems.
The book 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, applications and advanced topics are developed. One particularly notable application is to complex algebraic geometry, where important results from projective differential geometry are expanded and updated. 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. It has numerous exercises and examples throughout. The book will also be of use to experts in such areas as PDEs and algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields.
Graduate students and research mathematicians interested in differential geometry and exterior differential systems.
Table of Contents
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- Chapter 1. Moving frames and exterior differential systems
- Chapter 2. Euclidean geometry and Riemannian geometry
- Chapter 3. Projective geometry
- Chapter 4. Cartan-Kähler I: Linear algebra and constant-coefficient homogeneous systems
- Chapter 5. Cartan-Kähler II: The Cartan algorithm for linear Pfaffian systems
- Chapter 6. Applications to PDE
- Chapter 7. Cartan-Kähler III: The general case
- Chapter 8. Geometric structures and connections
- Appendix A. Linear algebra and representation theory
- Appendix B. Differential forms
- Appendix C. Complex structures and complex manifolds
- Appendix D. Initial value problems
- Hints and answers to selected exercises