Cartan for beginners : differential geometry via moving frames and exterior differential systems
著者
書誌事項
Cartan for beginners : differential geometry via moving frames and exterior differential systems
（Graduate studies in mathematics, v. 175）
American Mathematical Society, c2016
2nd ed
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注記
Includes bibliographical references (p. 437-444) and index
内容説明・目次
内容説明
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.
目次
Moving frames and exterior differential systems
Euclidean geometry
Riemannian geometry
Projective geometry I: Basic definitions and examples
Cartan-Kahler I: Linear algebra and constant-coefficient homogeneous systems
Cartan-Kahler II: The Cartan algorithm for linear Pfaffian systems
Applications to PDE
Cartan-Kahler III: The general case
Geometric structures and connections
Superposition for Darboux-integrable systems
Conformal differential geometry
Projective geometry II: Moving frames and subvarieties of projective space
Linear algebra and representation theory
Differential forms
Complex structures and complex manifolds
Initial value problems and the Cauchy-Kowalevski theorem
Hints and answers to selected exercises
Bibliography
Index.
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