Hardcover ISBN:  9781470409869 
Product Code:  GSM/175 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Electronic ISBN:  9781470435912 
Product Code:  GSM/175.E 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 

Book DetailsGraduate Studies in MathematicsVolume: 175; 2016; 455 ppMSC: Primary 35; 37; 53; 58;
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 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 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 Darbouxintegrable systems; and on conformal geometry, written in a manner to introduce readers to the related parabolic geometry perspective.ReadershipGraduate 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

Chapters

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. CartanKähler I: Linear algebra and constantcoefficient homogeneous systems

Chapter 6. CartanKähler II: The Cartan algorithm for linear Pfaffian systems

Chapter 7. Applications to PDE

Chapter 8. CartanKähler III: The general case

Chapter 9. Geometric structures and connections

Chapter 10. Superposition for Darbouxintegrable 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 CauchyKowalevski theorem


Additional Material

Reviews

[T]his book, like the first edition, is an excellent source for graduate students and professional mathematicians who want to learn about moving frames and Gstructures in trying to understand differential geometry.
Thomas Garrity, Mathematical Reviews 
All the material is carefully developed, many examples supporting the understanding. The reviewer warmly recommends this volume to mathematical university libraries.
Gabriel Eduard Vilcu, Zentralblatt MATH


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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 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 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 Darbouxintegrable 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

Chapters

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. CartanKähler I: Linear algebra and constantcoefficient homogeneous systems

Chapter 6. CartanKähler II: The Cartan algorithm for linear Pfaffian systems

Chapter 7. Applications to PDE

Chapter 8. CartanKähler III: The general case

Chapter 9. Geometric structures and connections

Chapter 10. Superposition for Darbouxintegrable 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 CauchyKowalevski theorem

[T]his book, like the first edition, is an excellent source for graduate students and professional mathematicians who want to learn about moving frames and Gstructures in trying to understand differential geometry.
Thomas Garrity, Mathematical Reviews 
All the material is carefully developed, many examples supporting the understanding. The reviewer warmly recommends this volume to mathematical university libraries.
Gabriel Eduard Vilcu, Zentralblatt MATH