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8 - Lectures on Moving Frames

Published online by Cambridge University Press:  05 July 2011

By
Peter J. Olver
Edited by
Decio Levi ,
Peter Olver ,
Zora Thomova  and
Pavel Winternitz
Peter J. Olver
Affiliation:
University of Minnesota
Decio Levi
Affiliation:
Università degli Studi Roma Tre
Peter Olver
Affiliation:
University of Minnesota
Zora Thomova
Affiliation:
SUNY Institute of Technology
Pavel Winternitz
Affiliation:
Université de Montréal
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Summary

Abstract

This chapter presents the equivariant method of moving frames for finite-dimensional Lie group actions, surveying a variety of applications, including geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant flows.

Introduction

According to Akivis [1], the method of moving frames originates in work of the Estonian mathematician Martin Bartels (1769–1836), a teacher of both Gauss and Lobachevsky. The field is most closely associated with Élie Cartan [22], who forged earlier contributions by Darboux, Frenet, Serret, and Cotton into a powerful tool for analyzing the geometric properties of submanifolds and their invariants under the action of transformation groups. In the 1970s, several researchers, cf. [25, 37, 38, 49], began the process of developing a firm theoretical foundation for the method. The final crucial step [32], is to define a moving frame simply as an equivariant map from the manifold back to the transformation group. All classical moving frames can be reinterpreted in this manner. Moreover, the equivariant approach is completely algorithmic, and applies to very general group actions.

Cartan's normalization construction of a moving frame can be interpreted as the choice of a cross-section to the group orbits. This enables one to algorithmically construct an equivariant moving frame along with a complete systems of invariants through the induced invariantization process. The existence of an equivariant moving frame requires freeness of the underlying group action, i.e., the isotropy subgroup of any single point is trivial.

Type
Chapter
Information
Symmetries and Integrability of Difference Equations , pp. 207 - 246
Publisher: Cambridge University Press
Print publication year: 2011

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