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

Marí Beffa, Gloria

Annales de l'Institut Fourier, Tome 58 (2008) no. 4, pp. 1295-1335.

Nous décrivons les repères mobiles et les invariants différentiels pour les courbes dans deux variétés paraboliques $G/H$, $G=O(p+1,q+1)$ et $G=O(2m,2m)$ et introduisons les invariants différentiels de type projectif. Dans le cas $G=O(p+1,q+1)$ nous montrons l’existence de flots géométriques sur $G/H$ qui induisent des équations de type KdV pour les invariants de type projectif (si les conditions initiales sont bien choisies). Nous montrons par ailleurs que le crochet de Poisson dans l’espace des invariants différentiels des courbes de $G/H$ peuvent être réduits à la sous-variété des invariants de type projectif où ils deviennent alors des structures Hamiltoniennes de type KdV. Dans le cas $G=O(2m,2m)$, nous classifions les invariants différentiels et montrons que, pour quelques repères mobiles bien choisis, il y a des flots géométriques sur $G/H$ qui induisent un système d’équations de KdV decouplé pour les invariants de type projectif, si les conditions initiales sont bien choisies. Nous détaillons la différence entre ce cas et le cas de la Grassmannienne Langrangienne.

In this paper we describe moving frames and differential invariants for curves in two different $\left|1\right|$-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.

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