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

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Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds AN N A L E S D E L’INSTI T U T F O U R IE R ANNALES DE L’INSTITUT FOURIER Gloria MARí BEFFA Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds Tome 58, no 4 (2008), p. 1295-1335. <http://aif.cedram.org/item?id=AIF_2008__58_4_1295_0> © Association des Annales de l’institut Fourier, 2008, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 58, 4 (2008) 1295-1335 PROJECTIVE-TYPE DIFFERENTIAL INVARIANTS AND GEOMETRIC CURVE EVOLUTIONS OF KDV-TYPE IN FLAT HOMOGENEOUS MANIFOLDS by Gloria MARí BEFFA Abstract. — In this paper we describe moving frames and differential in- variants 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 O

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