From Frenet To Cartan The Method Of Moving Frames
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From Frenet to Cartan: The Method of Moving Frames
Author: Jeanne N. Clelland
language: en
Publisher: American Mathematical Soc.
Release Date: 2017-03-29
The method of moving frames originated in the early nineteenth century with the notion of the Frenet frame along a curve in Euclidean space. Later, Darboux expanded this idea to the study of surfaces. The method was brought to its full power in the early twentieth century by Elie Cartan, and its development continues today with the work of Fels, Olver, and others. This book is an introduction to the method of moving frames as developed by Cartan, at a level suitable for beginning graduate students familiar with the geometry of curves and surfaces in Euclidean space. The main focus is on the use of this method to compute local geometric invariants for curves and surfaces in various 3-dimensional homogeneous spaces, including Euclidean, Minkowski, equi-affine, and projective spaces. Later chapters include applications to several classical problems in differential geometry, as well as an introduction to the nonhomogeneous case via moving frames on Riemannian manifolds. The book is written in a reader-friendly style, building on already familiar concepts from curves and surfaces in Euclidean space. A special feature of this book is the inclusion of detailed guidance regarding the use of the computer algebra system Maple™ to perform many of the computations involved in the exercises.
Cartan for Beginners
Author: Thomas Andrew Ivey
language: en
Publisher: American Mathematical Soc.
Release Date: 2003
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.
Riemannian Geometry in an Orthogonal Frame
Elie Cartan's book Geometry of Riemannian Manifolds (1928) was one of the best introductions to his methods. It was based on lectures given by the author at the Sorbonne in the academic year 1925-26. A modernized and extensively augmented edition appeared in 1946 (2nd printing, 1951, and 3rd printing, 1988). Cartan's lectures in 1926-27 were different -- he introduced exterior forms at the very beginning and used extensively orthonormal frames throughout to investigate the geometry of Riemannian manifolds. In this course he solved a series of problems in Euclidean and non-Euclidean spaces, as well as a series of variational problems on geodesics. The lectures were translated into Russian in the book Riemannian Geometry in an Orthogonal Frame (1960). This book has many innovations, such as the notion of intrinsic normal differentiation and the Gaussian torsion of a submanifold in a Euclidean multidimensional space or in a space of constant curvature, an affine connection defined in a normal fiber bundle of a submanifold, etc. The only book of Elie Cartan that was not available in English, it has now been translated into English by Vladislav V Goldberg, the editor of the Russian edition.