Jacobi Equations and Comparison Theorems for Corank 1 sub-Riemannian Structures with Symmetries

TL;DR

This paper develops a framework for analyzing Jacobi equations in corank 1 sub-Riemannian structures with symmetries, linking curvature maps to Riemannian curvature and magnetic fields, and estimating conjugate points.

Contribution

It provides explicit formulas for curvature maps in sub-Riemannian structures with symmetries, extending previous invariants to new geometric settings.

Findings

Explicit curvature map formulas in terms of Riemannian curvature and magnetic fields.

Estimates for conjugate points based on curvature bounds.

Application of the framework to structures in Yang-Mills fields.

Abstract

The Jacobi curve of an extremal of optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. In our previous works we constructed the canonical bundle of moving frames and the complete system of symplectic invariants, called curvature maps, for parametrized curves in Lagrange Grassmannians satisfying very general assumptions. The structural equation for a canonical moving frame of the Jacobi curve of an extremal can be interpreted as the normal form for the Jacobi equation along this extremal and the curvature maps can be seen as the "coefficients" of this normal form. In the case of a Riemannian metric there is only one curvature map and it is naturally related to the Riemannian sectional curvature. In the present paper we study the curvature maps for a sub-Riemannian structure on a corank 1 distribution having an additional transversal infinitesimal symmetry. After the factorization by the integral foliation of this symmetry, such sub-Riemannian structure can be reduced to a Riemannian manifold equipped with a closed 2-form (a magnetic field). We obtain explicit expressions for the curvature maps of the original sub-Riemannian structure in terms of the curvature tensor of this Riemannian manifold and the magnetic field. We also estimate the number of conjugate points along the sub-Riemannian extremals in terms of the bounds for the curvature tensor of this Riemannian manifold and the magnetic field in the case of an uniform magnetic field. The language developed for the calculation of the curvature maps can be applied to more general sub-Riemannian structures with symmetries, including sub-Riemmannian structures appearing naturally in Yang-Mills fields.