Parametrized curves in Lagrange Grassmannians

Igor Zelenko; Chengbo Li

doi:10.1016/j.crma.2007.10.034

Differential Geometry

Parametrized curves in Lagrange Grassmannians
[Courbes paramétrées dans les grassmanniennes lagrangiennes]

Présenté par : 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.

Résumé
Abstract

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.

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.

Reçu le : 2007-08-08
Accepté le : 2007-10-16
Publié le : 2007-11-19

DOI : 10.1016/j.crma.2007.10.034

Affiliations des auteurs :

Igor Zelenko ¹ ; Chengbo Li ¹

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

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     title = {Parametrized curves in {Lagrange} {Grassmannians}},
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     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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.034/

Bibliographie
Cité par

[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

Chengbo Li Curvatures and hyperbolic flows for natural mechanical systems in Finsler geometry, Journal of Dynamical and Control Systems, Volume 28 (2022) no. 2, pp. 249-266 | DOI:10.1007/s10883-021-09533-6 | Zbl:1492.53046
Andrei Agrachev; Davide Barilari; Ugo Boscain A Comprehensive Introduction to Sub-Riemannian Geometry, 2019 | DOI:10.1017/9781108677325
Frédéric Jean; Sofya Maslovskaya; Igor Zelenko On projective and affine equivalence of sub-Riemannian metrics, Geometriae Dedicata, Volume 203 (2019), pp. 279-319 | DOI:10.1007/s10711-019-00437-1 | Zbl:1475.53035
Mauricio Godoy Molina; Erlend Grong Riemannian and sub-Riemannian geodesic flows, The Journal of Geometric Analysis, Volume 27 (2017) no. 2, pp. 1260-1273 | DOI:10.1007/s12220-016-9717-8 | Zbl:1380.53037
Carlos E. Durán; Cíntia R. A. Peixoto Geometry of fanning curves in divisible Grassmannians, Differential Geometry and its Applications, Volume 49 (2016), pp. 447-472 | DOI:10.1016/j.difgeo.2016.10.001 | Zbl:1352.53012
Paul W. Y. Lee; Chengbo Li; Igor Zelenko Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds., Discrete and Continuous Dynamical Systems, Volume 36 (2016) no. 1, pp. 303-321 | DOI:10.3934/dcds.2016.36.303 | Zbl:1326.53043
Erlend Grong; Anton Thalmaier Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations. I, Mathematische Zeitschrift, Volume 282 (2016) no. 1-2, pp. 99-130 | DOI:10.1007/s00209-015-1534-4 | Zbl:1361.53028
Andrei Agrachev; Paul W. Y. Lee Bishop and Laplacian comparison theorems on three-dimensional contact sub-Riemannian manifolds with symmetry, The Journal of Geometric Analysis, Volume 25 (2015) no. 1, pp. 512-535 | DOI:10.1007/s12220-013-9437-2 | Zbl:1341.53058
Chengbo Li On curvature-type invariants for natural mechanical systems on sub-Riemannian structures associated with a principle G-bundle, Geometric Control Theory and Sub-Riemannian Geometry, Volume 5 (2014), p. 263 | DOI:10.1007/978-3-319-02132-4_16
Andrei Agrachev; Paul W. Y. Lee Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds, Mathematische Annalen, Volume 360 (2014) no. 1-2, pp. 209-253 | DOI:10.1007/s00208-014-1034-6 | Zbl:1327.53032
Boris Doubrov; Igor Zelenko On geometry of curves of flags of constant type, Central European Journal of Mathematics, Volume 10 (2012) no. 5, pp. 1836-1871 | DOI:10.2478/s11533-012-0078-7 | Zbl:1262.53013
Chengbo Li; Igor Zelenko Jacobi equations and comparison theorems for corank 1 sub-Riemannian structures with symmetries, Journal of Geometry and Physics, Volume 61 (2011) no. 4, pp. 781-807 | DOI:10.1016/j.geomphys.2010.12.009 | Zbl:1216.53039
Igor Zelenko; Chengbo Li Differential geometry of curves in Lagrange Grassmannians with given Young diagram, Differential Geometry and its Applications, Volume 27 (2009) no. 6, pp. 723-742 | DOI:10.1016/j.difgeo.2009.07.002 | Zbl:1177.53020

Cité par 13 documents. Sources : Crossref, zbMATH

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