Description
Book SynopsisOffers 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.
Trade ReviewAn 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
Table of Contents
- 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.
