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Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Second Edition

Thomas A. Ivey College of Charleston, Charleston, SC
Joseph M. Landsberg Texas A&M University, College Station, TX
Available Formats:
Hardcover ISBN: 978-1-4704-0986-9
Product Code: GSM/175
List Price: $89.00 MAA Member Price:$80.10
AMS Member Price: $71.20 Electronic ISBN: 978-1-4704-3591-2 Product Code: GSM/175.E List Price:$89.00
MAA Member Price: $80.10 AMS Member Price:$71.20
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List Price: $133.50 MAA Member Price:$120.15
AMS Member Price: $106.80 Click above image for expanded view Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Second Edition Thomas A. Ivey College of Charleston, Charleston, SC Joseph M. Landsberg Texas A&M University, College Station, TX Available Formats:  Hardcover ISBN: 978-1-4704-0986-9 Product Code: GSM/175  List Price:$89.00 MAA Member Price: $80.10 AMS Member Price:$71.20
 Electronic ISBN: 978-1-4704-3591-2 Product Code: GSM/175.E
 List Price: $89.00 MAA Member Price:$80.10 AMS Member Price: $71.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$133.50 MAA Member Price: $120.15 AMS Member Price:$106.80
• Book Details

Volume: 1752016; 455 pp
MSC: 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 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

• 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. 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

• 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 G-structures 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1752016; 455 pp
MSC: 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 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

• 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. 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
• [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 G-structures 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
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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