Parametrized curves in Lagrange Grassmannians

Igor Zelenko; Chengbo Li

doi:10.1016/j.crma.2007.10.034

Differential Geometry

Parametrized curves in Lagrange Grassmannians

Presented by: Pierre Deligne

Igor Zelenko ¹ ; Chengbo Li ¹

¹ S.I.S.S.A., Via Beirut 2-4, 34014 Trieste, Italy

Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 647-652

Abstract (Original language)
Résumé

Curves in Lagrange Grassmannians naturally appear when one studies Jacobi equations for extremals, associated with geometric structures on manifolds. We fix integers $d_{i}$ and consider curves $Λ (t)$ for which at each t the derivatives of order ⩽i of all curves of vectors $ℓ (t) \in Λ (t)$ span a subspace of dimension $d_{i}$ . We will describe the construction of a complete system of symplectic invariants for such parametrized curves, satisfying a certain genericity assumption, and give applications to geometric structures, including sub-Riemannian and sub-Finslerian structures.

Les courbes dans les grassmanniennes lagrangiennes apparaissent naturellement lors de l'étude intrinsèque des « équations de Jacobi pour les extremas », associées à des structures géométriques sur les variétés différentielles. Nous fixons des entiers $d_{i}$ et considérons les courbes $Λ (t)$ pour lesquelles en chaque t les dérivées d'ordre ⩽i des $ℓ (t) \in Λ (t)$ engendrent un sous-espace de dimension $d_{i}$ . Nous décrirons la construction d'un système complet d'invariants symplectiques pour de telles courbes paramétrées vérifiant une condition de généricité, et nous donnerons des applications à la géométrie différentielle de structures géométriques, incluant les structures sous-riemanniennes et sous-finsleriennes.

Received: 2007-08-08
Accepted: 2007-10-16
Published online: 2007-11-19

DOI: 10.1016/j.crma.2007.10.034

Author's affiliations:

Igor Zelenko ¹ ; Chengbo Li ¹

¹ S.I.S.S.A., Via Beirut 2-4, 34014 Trieste, Italy

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     author = {Igor Zelenko and Chengbo Li},
     title = {Parametrized curves in {Lagrange} {Grassmannians}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {647--652},
     year = {2007},
     publisher = {Elsevier},
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     number = {11},
     doi = {10.1016/j.crma.2007.10.034},
     language = {en},
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Igor Zelenko; Chengbo Li. Parametrized curves in Lagrange Grassmannians. Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 647-652. doi: 10.1016/j.crma.2007.10.034

References
Cited by

[1] A. Agrachev; I. Zelenko Geometry of Jacobi curves. I, J. Dynam. Control Systems, Volume 8 (2002) no. 1, pp. 93-140

[2] I. Zelenko Complete systems of invariants for rank 1 curves in Lagrange Grassmannians, Differential Geometry and its Applications, Proc. Conf. Prague, August 30–September 3, 2004, Charles University, Prague, 2005, pp. 365-379

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