12 - Hörmander Manifolds

Summary

Definition of Hörmander Manifolds

The step of a sub-Riemannian manifold (M, D) at a point p is equal to 1 plus the maximum number of iterations of the Lie brackets of horizontal vector fields needed to be taken to generate the tangent space T_pM. The sub-Riemannian manifolds with step 2 everywhere correspond to Heisenberg manifolds, while those with the constant step 1 correspond to Riemannian manifolds. Any manifold that has the step greater than or equal to 3 at one or more points falls into a new category of sub-Riemannian manifolds, which we shall call Hörmander manifolds.

We have the following definition similar with the one about Heisenberg manifolds.

Definition 12.1.1.A Hörmander manifold is a sub-Riemannian manifold (M, D, g) such that:

1. • the distribution D is bracket generating, with points where the step is at least 3

2. • there are k, k < dim M, locally defined horizontal vector fields on M, such that

3. g(X_i, X_j) = δ_ijand D_p = span{X₁,…, X_k}_p, for all p ∈ M.

On a Hörmander manifold the Lagrangian and the Hamiltonian formalisms are no more equivalent. In this case we make the distinction between the geodesics obtained by one or the other formalisms.