The Schouten curvature and the Jacobi equation in sub-Riemannian geometry

We show that if a distribution does not depend on the vertical coordinates, then the Schouten curvature tensor coincides with the Riemannian curvature. The Schouten curvature tensor and the nonholonomicity tensor are used to write the Jacobi equation for the distribution. This leads to a study of second-order optimality conditions for horizontal geodesics in sub-Riemannian geometry. We study conjugate points for horizontal geodesics on the Heisenberg group as an example.