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Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems
Thomas A. Ivey, College of Charleston, SC, and J.M. Landsberg, Georgia Institute of Technology, Atlanta, GA

Graduate Studies in Mathematics
2003; 378 pp; hardcover
Volume: 61
ISBN-10: 0-8218-3375-8
ISBN-13: 978-0-8218-3375-9
List Price: US$65
Member Price: US$52
Order Code: GSM/61
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See also:

Curves and Surfaces: Second Edition - Sebastian Montiel and Antonio Ros

Tensors: Geometry and Applications - J M Landsberg

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 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 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

  • Moving frames and exterior differential systems
  • Euclidean geometry and Riemannian geometry
  • Projective geometry
  • Cartan-Kähler I: Linear algebra and constant-coefficient homogeneous systems
  • Cartan-Kähler II: The Cartan algorithm for linear Pfaffian systems
  • Applications to PDE
  • Cartan-Kähler III: The general case
  • Geometric structures and connections
  • Linear algebra and representation theory
  • Differential forms
  • Complex structures and complex manifolds
  • Initial value problems
  • Hints and answers to selected exercises
  • Bibliography
  • Index
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