exists a Hamiltonian, called the sub-Riemannian Hamiltonian, constructed out of the metric for the manifold. Conversely, every such quadratic Hamiltonian induces a sub-Riemannian manifold. The existence of geodesics of the corresponding Hamilton–Jacobi equations for the sub-Riemannian Hamiltonian is given by the Chow–Rashevskii theorem. See also Carnot group, a class of Lie groups that form sub-Riemannian manifolds Distribution References Metric geometry Riemannian geometry Riemannian manifolds