Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds

Gloria Marí Beffa[1]

  • [1] University of Wisconsin Mathematics Department Madison, Wisconsin 53706 (USA)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 4, page 1295-1335
  • ISSN: 0373-0956

Abstract

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In this paper we describe moving frames and differential invariants for curves in two different | 1 | -graded parabolic manifolds G / H , G = O ( p + 1 , q + 1 ) and G = O ( 2 m , 2 m ) , and we define differential invariants of projective-type. We then show that, in the first case, there are geometric flows in G / H inducing equations of KdV-type in the projective-type differential invariants when proper initial conditions are chosen. We also show that geometric Poisson brackets in the space of differential invariants of curves in G / H can be reduced to the submanifold of invariants of projective-type to become Hamiltonian structures of KdV-type. The study is based on the use of Fels and Olver moving frames. In the second case we classify differential invariants and we show that for some choices of moving frames we can find geometric evolutions inducing a decoupled system of KdV equations on the projective-type differential invariants, if proper initial values are chosen. We describe the differences between this case and the Lagrangian Grassmannian case in detail.

How to cite

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Marí Beffa, Gloria. "Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds." Annales de l’institut Fourier 58.4 (2008): 1295-1335. <http://eudml.org/doc/10349>.

@article{Marí2008,
abstract = {In this paper we describe moving frames and differential invariants for curves in two different $|1|$-graded parabolic manifolds $G/H$, $G = O(p+1,q+1)$ and $G = O(2m,2m)$, and we define differential invariants of projective-type. We then show that, in the first case, there are geometric flows in $G/H$ inducing equations of KdV-type in the projective-type differential invariants when proper initial conditions are chosen. We also show that geometric Poisson brackets in the space of differential invariants of curves in $G/H$ can be reduced to the submanifold of invariants of projective-type to become Hamiltonian structures of KdV-type. The study is based on the use of Fels and Olver moving frames. In the second case we classify differential invariants and we show that for some choices of moving frames we can find geometric evolutions inducing a decoupled system of KdV equations on the projective-type differential invariants, if proper initial values are chosen. We describe the differences between this case and the Lagrangian Grassmannian case in detail.},
affiliation = {University of Wisconsin Mathematics Department Madison, Wisconsin 53706 (USA)},
author = {Marí Beffa, Gloria},
journal = {Annales de l’institut Fourier},
keywords = {Invariant evolutions of curves; flat homogeneous spaces; Poisson brackets; differential invariants; projective invariants; completely integrable PDEs; moving frames; invariant evolutions of curves},
language = {eng},
number = {4},
pages = {1295-1335},
publisher = {Association des Annales de l’institut Fourier},
title = {Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds},
url = {http://eudml.org/doc/10349},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Marí Beffa, Gloria
TI - Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 4
SP - 1295
EP - 1335
AB - In this paper we describe moving frames and differential invariants for curves in two different $|1|$-graded parabolic manifolds $G/H$, $G = O(p+1,q+1)$ and $G = O(2m,2m)$, and we define differential invariants of projective-type. We then show that, in the first case, there are geometric flows in $G/H$ inducing equations of KdV-type in the projective-type differential invariants when proper initial conditions are chosen. We also show that geometric Poisson brackets in the space of differential invariants of curves in $G/H$ can be reduced to the submanifold of invariants of projective-type to become Hamiltonian structures of KdV-type. The study is based on the use of Fels and Olver moving frames. In the second case we classify differential invariants and we show that for some choices of moving frames we can find geometric evolutions inducing a decoupled system of KdV equations on the projective-type differential invariants, if proper initial values are chosen. We describe the differences between this case and the Lagrangian Grassmannian case in detail.
LA - eng
KW - Invariant evolutions of curves; flat homogeneous spaces; Poisson brackets; differential invariants; projective invariants; completely integrable PDEs; moving frames; invariant evolutions of curves
UR - http://eudml.org/doc/10349
ER -

References

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