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.

Classification : 37Kxx, 53A55
Keywords: Invariant evolutions of curves, flat homogeneous spaces, Poisson brackets, differential invariants, projective invariants, completely integrable PDEs, moving frames.
Mot clés : repères mobiles, invariants différentiels de type projectif, équations de type KdV, structures Hamiltoniennes de type KdV.

[1] Calini, A. Recent developments in integrable curve dynamics, Geometric approaches to differential equations (Canberra 1995) (2000), pp. 56-99 (Cambridge University Press, Cambridge) | MR | Zbl

[2] Cartan, E. La Méthode du Repère Mobile, la Théorie des Groupes Continus et les Espaces Généralisés, Exposés de Géométrie 5, Hermann, Paris, 1935 | Zbl

[3] Fels, M.; Olver, P. J. Moving coframes. I. A practical algorithm, Acta Appl. Math. (1997), pp. 99-136 | MR | Zbl

[4] Fels, M.; Olver, P. J. Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. (1999), pp. 127-208 | DOI | MR | Zbl

[5] Fialkow, A. The Conformal Theory or Curves, Transactions of the AMS, Volume 51 (1942), pp. 435-568 | MR | Zbl

[6] Green, M. L. The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces, Duke Mathematical Journal, Volume 45 (1978) no. 4, pp. 735-779 | DOI | MR | Zbl

[7] Griffiths, P. A. On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in Differential Geometry, Duke Mathematical Journal, Volume 41 (1974), pp. 775-814 | DOI | Zbl

[8] Hasimoto, R. A soliton on a vortex filament, J. Fluid Mechanics, Volume 51 (1972), pp. 477-485 | DOI | Zbl

[9] Hubert, E. Generation properties of differential invariants in the moving frame methods (in preparation)

[10] Kobayashi, S. Transformation Groups in Differential Geometry, Classics in Mathematics, Springer–Verlag, New York, 1972 | MR | Zbl

[11] Kobayashi, S.; Nagano, T. On filtered Lie Algebras and Geometric Structures I, Journal of Mathematics and Mechanics, Volume 13 (1964) no. 5, pp. 875-907 | MR | Zbl

[12] Marí Beffa, G. Geometric Poisson brackets in flat semisimple homogenous spaces (accepted for publication in the Asian Journal of Mathematics.)

[13] Marí Beffa, G. Poisson brackets associated to the conformal geometry of curves, Trans. Amer. Math. Soc., Volume 357 (2005), pp. 2799-2827 | DOI | MR | Zbl

[14] Marí Beffa, G. Poisson Geometry of differential invariants of curves in nonsemisimple homogeneous spaces, Proc. Amer. Math. Soc., Volume 134 (2006), pp. 779-791 | DOI | MR | Zbl

[15] Marí Beffa, G. On completely integrable geometric evolutions of curves of Lagrangian planes, Proceedings of the Royal academy of Edinburg, Volume 137A (2007), pp. 111-131 | MR | Zbl

[16] Ochiai, T. Geometry associated with semisimple flat homogeneous spaces, Transactions of the AMS, Volume 152 (1970), pp. 159-193 | DOI | MR | Zbl

[17] Olver, P. J. Equivalence, Invariance and Symmetry, Cambridge University Press, Cambridge, UK, 1995 | MR | Zbl

[18] Olver, P. J. Moving frames and singularities of prolonged group actions, Selecta Math, Volume 6 (2000), pp. 41-77 | DOI | MR | Zbl

[19] Olver, P. J.; Wang, J. P. Classification of one-component systems on associative algebras, Proc. London Math. Soc., Volume 81 (2000), pp. 566-586 | DOI | MR | Zbl

[20] Ovsiannikov, L. V. Group Analysis of Differential Equations, Academic Press, New York, 1982 | MR | Zbl

[21] Ovsienko, V. Lagrange Schwarzian derivative and symplectic Sturm theory, Ann. de la Fac. des Sciences de Toulouse, Volume 6 (1993) no. 2, pp. 73-96 | DOI | Numdam | MR | Zbl

[22] Ovsienko, V.; Tabachnikov, S. Projective differential geometry, old and new, Cambridge tracts in Mathematics, Cambridge University press, 2005 | MR | Zbl

[23] Williamson, J. Normal matrices over an arbitrary field of characteristic zero, American Journal of Mathematics, Volume 61 (1939) no. 2, pp. 335-356 | DOI | MR