Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds
type: Item
instance of: scholarly article
title: Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds
zbMATH Open document ID: 1192.37099
author: G. Marí Beffa
publication date: 2008-07-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10349
review text: A new concept of moving frame was introduced by Fels and Olver in 1999--2000. Instead of the classical concept of moving frame along a curve, they looked at the moving frame as an equivariant map. A fundamental problem with the traditional moving frame was that in non-affine geometries, the classical Serret-Frenet equation did not provide a complete set of generators of differential invariants for the curve. In the new setting, however, a complete set of generators can always be found among the entries of the matrix defining its Serret-Frenet equations. In this paper, the author describes moving frames and differential invariants for curves in two different parabolic manifolds \(G/H\), where \(G= O(p+1,q+1)\) and \(G= O(2m,2m)\). He shows that, in the first case, there are geometric flows inducing equations of the KdV type in the projective-type differential invariants when proper initial conditions are chosen. In the second case, he classifies differential invariants and shows that for some choices of moving frames, one can find geometric evolutions inducing a decoupled system of KdV equations on the projective-type differential invariants. The differences between this case and the Lagrangian Grassmannian case are also described in detail.
Mathematics Subject Classification ID: 37K25
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 5303676
zbMATH Keywords: invariant evolutions of curves
zbMATH Keywords: flat homogeneous spaces
zbMATH Keywords: Poisson brackets
zbMATH Keywords: differential invariants
zbMATH Keywords: projective invariants
zbMATH Keywords: completely integrable PDEs
zbMATH Keywords: moving frames
Wikidata QID: Q115159034
reviewed by: Dmitry Pelinovsky
MaRDI profile type: Publication
OpenAlex ID: W2492783678
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DOI: 10.5802/AIF.2385
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5303676