The Jacobi equation in sub-Riemannian geometry

Non-holonomic distributions are the most natural generalization of smooth manifolds. Distribution is a field of k-planes on an n-manifold, k<n, which smoothly depends on the point of a manifold. The sub-Riemannian connection and curvature are well defined on distributions. The geodesics equation depends not only on the connection but also on the non-holonomicity (tensions, or torsion) tensor. Hence the geodesic flow is quite different from the Riemannian geometry. We consider the attached variational problem and write the Jacobi equation for the Heisenberg group.