
Book DetailsGraduate Studies in MathematicsVolume: 61; 2003; 378 ppMSC: Primary 53; Secondary 58; 35; 14
Now available in Second Edition: GSM/175
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 oneyear graduate course in differential geometry, and parts of it can be used for a onesemester 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.
ReadershipGraduate students and research mathematicians interested in differential geometry and exterior differential systems.

Table of Contents

Chapters

Chapter 1. Moving frames and exterior differential systems

Chapter 2. Euclidean geometry and Riemannian geometry

Chapter 3. Projective geometry

Chapter 4. CartanKähler I: Linear algebra and constantcoefficient homogeneous systems

Chapter 5. CartanKähler II: The Cartan algorithm for linear Pfaffian systems

Chapter 6. Applications to PDE

Chapter 7. CartanKähler 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


Additional Material

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Now available in Second Edition: GSM/175
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 oneyear graduate course in differential geometry, and parts of it can be used for a onesemester 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.
Graduate students and research mathematicians interested in differential geometry and exterior differential systems.

Chapters

Chapter 1. Moving frames and exterior differential systems

Chapter 2. Euclidean geometry and Riemannian geometry

Chapter 3. Projective geometry

Chapter 4. CartanKähler I: Linear algebra and constantcoefficient homogeneous systems

Chapter 5. CartanKähler II: The Cartan algorithm for linear Pfaffian systems

Chapter 6. Applications to PDE

Chapter 7. CartanKähler 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