Cartan for beginners : differential geometry via moving frames and exterior differential systems / Thomas A. Ivey, J. M. LandsbergType de document : MonographieCollection : Graduate studies in mathematics, 61Langue : anglais.Pays : Etats Unis.Éditeur : Providence : American Mathematical Society, 2003Description : 1 vol. (XIV-378 p.) : ill. ; 26 cmISBN : 9780821833759.ISSN : 1065-7339.Bibliographie : Bibliogr. p. 363-369. Index.Sujet MSC : 53A20, Differential geometry -- Classical differential geometry, Projective differential geometry
58A15, Global analysis, analysis on manifolds -- General theory of differentiable manifolds, Exterior differential systems (Cartan theory)
53C10, Differential geometry -- Global differential geometry, G-structures
53C05, Differential geometry -- Global differential geometry, Connections, general theory
34C41, Ordinary differential equations -- Qualitative theory, Equivalence, asymptotic equivalenceEn-ligne : Zentralblatt | MathSciNet | AMS
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The purpose of this book is to provide a systematic and self-contained introduction to the method of moving frames and to the theory of exterior differential systems. Thanks to its lively expository style, the well-balanced selection of topics and the use of many interesting examples and problems ranging from classical to modern, this book can make an excellent "first reading'' in the subject and can be useful as a text for a graduate course. The material covered requires some knowledge of classical differential and algebraic geometry, differentiable manifolds and Lie groups. On the other hand, researchers in areas such as PDE and algebraic geometry will find it a valuable reference to learn how moving frames and exterior differential systems apply to their fields.
Topics covered include: the method of moving frames as applied to the geometry of submanifolds of homogeneous spaces, and the rudiments of Riemannian geometry; an extensive study of the projective differential geometry of submanifolds with applications to algebraic geometry, including the Zak and Landman theorems on the dual defect, results on complete intersections, osculating hypersurfaces, uniruled varieties and varieties covered by lines; constant coefficient homogeneous systems of PDE's, tableaux and involutivity of tableaux; the Cartan algorithm for linear Pfaffian systems, the isometric embedding problem, calibrated submanifolds; the classical theory of characteristic, derived systems, Monge-Ampère systems, Weingarten surfaces, integrable extensions and Bäcklund transformations; the Cartan-Kähler theorem and Cartan's test for regularity; applications to geometric structures, connections, holonomy and path geometry in the plane.
For the reader's convenience, most of the required prerequisites are given in four appendices devoted to representation theory, exterior differential forms and vector fields, complex and almost complex manifolds and to the Cauchy-Kovalevskaya theorem. (MathSciNet)
Bibliogr. p. 363-369. Index