Differential geometry of curves in Lagrange Grassmannians with given Young diagram

TL;DR

This paper develops a comprehensive system of symplectic invariants for curves in Lagrange Grassmannians characterized by Young diagrams, enabling a unified local differential geometry approach for diverse geometric structures.

Contribution

It introduces a complete set of symplectic invariants for parameterized curves in Lagrange Grassmannians with specified Young diagrams, unifying geometric analysis across multiple structures.

Findings

Constructed symplectic invariants for curves with given Young diagrams.

Unified local differential geometry framework for classical and non-classical structures.

Applicable to Riemannian, Finslerian, sub-Riemannian, and sub-Finslerian geometries.

Abstract

Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric 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 optimal 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 symplectic 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 construction 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.