On the classification of Jacobi curves and their conformal curvatures

Abstract

This paper describes the theory of Jacobi curves, a far reaching extension of the spaces of Jacobi fields along Riemannian geodesics, developed by Agrachev and Zelenko. Jacobi curves are curves in the Lagrangian Grassmannian of a symplectic space satisfying appropriate regularity conditions. It is shown that they are fully characterised in terms of a family of conformal symplectic invariant curvatures. In addition to a new derivation of the Ricci curvature tensor of a Jacobi curve, a Cartan-like theory of Jacobi curves is presented that allows to associate to any admissible Jacobi curve a reduced normal Cartan matrix. A reconstruction theorem proving that an admissible Jacobi curve is characterised, up to conformal symplectic transformations, by a reduced normal Cartan matrix and a geometric parametrization is obtained. The theory of cycles is studied proving that they correspond to flat Jacobi curves.