From Frenet to Cartan

Description:

"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 MapleTM to perform many of the computations involved in the exercises"--amazon.com.

subjects: Vector analysis,  Geometry, differential,  Mathematical physics,  Frames (Vector analysis),  Exterior differential systems,  Differential Geometry,  Lie groups Topological groups,  Noncompact transformation groups,  Homogeneous spaces,  Classical differential geometry,  Curves in Euclidean space,  Surfaces in Euclidean space,  Affine differential geometry,  Projective differential geometry,  Differential invariants (local theory), geometric objects,  Local differential geometry,  Local submanifolds,  Lorentz metrics, indefinite metrics,  Global analysis, analysis on manifolds,  General theory of differentiable manifolds,  Differential forms,  Exterior differential systems (Cartan theory)