Principal invariants of Jacobi curves

Abstract

Jacobi curves are far going generalizations of the spaces of “Jacobi fields” along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. Differential geometry of these curves provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. In the present paper we mainly discuss two principal invariants: the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmannian providing the curve with a natural projective structure, and a fundamental form, which is a 4-order differential on the curve. This paper is a continuation of the works [1, 2], where Jacobi curves were defined, although it can be read independently.