Differential Geometry
Parametrized curves in Lagrange Grassmannians

Presented by: Pierre Deligne

Zelenko, Igor1; Li, Chengbo1
1 S.I.S.S.A., Via Beirut 2-4, 34014 Trieste, Italy
Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 647-652.

Curves in Lagrange Grassmannians naturally appear when one studies Jacobi equations for extremals, associated with geometric structures on manifolds. We fix integers di and consider curves Λ(t) for which at each t the derivatives of order ⩽i of all curves of vectors ℓ(t)∈Λ(t) span a subspace of dimension di. 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 di et considérons les courbes Λ(t) pour lesquelles en chaque t les dérivées d'ordre ⩽i des ℓ(t)∈Λ(t) engendrent un sous-espace de dimension di. 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:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.10.034
Zelenko, Igor 1; Li, Chengbo 1

1 S.I.S.S.A., Via Beirut 2-4, 34014 Trieste, Italy 
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Zelenko, Igor; Li, Chengbo. Parametrized curves in Lagrange Grassmannians. Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 647-652. doi : 10.1016/j.crma.2007.10.034. https://www.numdam.org/articles/10.1016/j.crma.2007.10.034/

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

[2] Zelenko, I. 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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