Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Second Edition
About this Title
Thomas A. Ivey, College of Charleston, Charleston, SC and Joseph M. Landsberg, Texas A&M University, College Station, TX
Publication: Graduate Studies in Mathematics
Publication Year: 2016; Volume 175
ISBNs: 978-1-4704-0986-9 (print); 978-1-4704-3591-2 (online)
MathSciNet review: MR2003610
MSC: Primary 53-01; Secondary 58A15
Two central aspects of Cartan's approach to differential geometry are the theory of exterior differential systems (EDS) 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 in geometry. It begins with the classical differential 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. As well, the book features an introduction to $G$-structures and a treatment of the theory of connections. The techniques of EDS are 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 geometry of PDE systems and complex algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields.
The second edition features three new chapters: on Riemannian geometry, emphasizing the use of representation theory; on the latest developments in the study of Darboux-integrable systems; and on conformal geometry, written in a manner to introduce readers to the related parabolic geometry perspective.
Graduate students and researchers interested in differential geometry, in particular, in exterior systems and the moving frames method and in its applications in algebraic geometry, PDE, and other areas of mathematics
Table of Contents
- Chapter 1. Moving frames and exterior differential systems
- Chapter 2. Euclidean geometry
- Chapter 3. Riemannian geometry
- Chapter 4. Projective geometry I: Basic definitions and examples
- Chapter 5. Cartan-Kähler I: Linear algebra and constant-coefficient homogeneous systems
- Chapter 6. Cartan-Kähler II: The Cartan algorithm for linear Pfaffian systems
- Chapter 7. Applications to PDE
- Chapter 8. Cartan-Kähler III: The general case
- Chapter 9. Geometric structures and connections
- Chapter 10. Superposition for Darboux-integrable systems
- Chapter 11. Conformal differential geometry
- Chapter 12. Projective geometry II: Moving frames and subvarieties of projective space
- Appendix A. Linear algebra and representation theory
- Appendix B. Differential forms
- Appendix C. Complex structures and complex manifolds
- Appendix D. Initial value problems and the Cauchy-Kowalevski theorem