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 (TM) to perform many of the computations involved in the exercises.
Rezensionen / Stimmen
An excellent and unique graduate level exposition of the differential geometry of curves, surfaces and higher-dimensional submanifolds of homogeneous spaces based on the powerful and elegant method of moving frames. The treatment is self-contained and illustrated through a large number of examples and exercises, augmented by Maple code to assist in both concrete calculations and plotting. Highly recommended." - Niky Kamran, McGill University
"The method of moving frames has seen a tremendous explosion of research activity in recent years, expanding into many new areas of applications, from computer vision to the calculus of variations to geometric partial differential equations to geometric numerical integration schemes to classical invariant theory to integrable systems to infinite-dimensional Lie pseudo-groups and beyond. Cartan theory remains a touchstone in modern differential geometry, and Clelland's book provides a fine new introduction that includes both classic and contemporary geometric developments and is supplemented by Maple symbolic software routines that enable the reader to both tackle the exercises and delve further into this fascinating and important field of contemporary mathematics. Recommended for students and researchers wishing to expand their geometric horizons." - Peter Olver, University of Minnesota
Reihe
Sprache
Verlagsort
Zielgruppe
Maße
Höhe: 254 mm
Breite: 178 mm
Gewicht
ISBN-13
978-1-4704-2952-2 (9781470429522)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Klassifikation
Jeanne N. Clelland, University of Colorado, Boulder, CO.
Background material: Assorted notions from differential geometry
Differential forms
Curves and surfaces in homogeneous spaces via the method of moving frames: Homogeneous spaces
Curves and surfaces in Euclidean space
Curves and surfaces in Minkowski space
Curves and surfaces in equi-affine space
Curves and surfaces in projective space
Applications of moving frames: Minimal surfaces in $\mathbb{E}^3$ and $\mathbb{A}^3$
Pseudospherical surfaces in Backlund's theorem
Two classical theorems
Beyond the flat case: Moving frames on Riemannian manifolds: Curves and surfaces in elliptic and hyperbolic spaces
The nonhomogeneous case: Moving frames on Riemannian manifolds
Bibliography
Index.
