Differential geometry of curves in Lagrange Grassmannians with given Young diagram

By Igor Zelenko, Chengbo Li. Differential geometry and its applications, 2007

84 citations recorded by the source corpus.

Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geo- metric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associate with a geometric structure. The Pontryagin extremals of this op- timal problem are integral curves of certain Hamiltonian system in the cotangent bundle. The dynamics of the fibers of the cotangent bundle w.r.t. this system along an extremal is described by certain curve in a Lagrange Grassmannian, called Jacobi curve of the extremal. Any sym- plectic invariant of the Jacobi curves produces the invariant of the original geometric structure. The basic characteristic of a curve in a Lagrange Grassmannian is its Young diagram. The number of boxes in its kth column is equal to the rank of the kth derivative of the curve (which is an appropriately defined linear mapping) at a generic point. We will describe the construc- tion of the complete system of symplectic invariants for parameterized curves in a Lagrange Grassmannian with given Young diagram. It allows to develop in a unified way local differential geometry of very wide classes of geometric structures on manifolds, including both classical geometric structures such as Riemannian and Finslerian structures and less classical ones such as sub-Riemannian and sub-Finslerian structures, defined on nonholonomic distributions.

Subfields: Astronomy and Astrophysics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/57196d7aa9755369dccdf89954bda4b389d54348

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In Astronomy and Astrophysics

1. Davide Barilari
2. L. Rizzi
3. Andrei A. Agrachev
4. Igor Zelenko
5. Ugo V. Boscain
6. R. B. Tully

In Statistical and Nonlinear Physics

1. Andrei A. Agrachev
2. Davide Barilari
3. L. Rizzi
4. Fabrice Baudoin
5. Nicola Garofalo

Citations in the corpus

• Sub-Riemannian Curvature in Contact Geometry (2015, methodology): “This is achieved by the so-called canonical Darboux frame, introduced in [5, 20,29].”