Sub-Riemannian Manifolds

Roughly speaking, a sub-Riemannian manifold is a manifold that has measuring restrictions. We are allowed to measure the magnitude of vectors only for a distinguished subset of vectors, called horizontal vectors. In other words, we cannot observe the manifold in all directions, fact that allows for “missing” or “forbidden” directions. The existence of these directions is intimately related to the noncommutativity properties built on the manifold. More precisely, we have the following definition.

Definition 2.1.1.A sub-Riemannian manifold is a real manifold M of dimension n together with a nonintegrable distribution D of rank k (k < n) endowed with a sub-Riemannian metric g, i.e., an application gp: Dp × Dp → ℝ, for all p ∈ M, which is a positive de?nite, nondegenerate, inner product.

We note that the rank k cannot be equal to the dimension n because in this case the distribution D will be tangent to the manifold M and then Dx = TxM; i.e., the distribution D becomes integrable because M is an integral manifold.

The nonintegrability condition of the distribution D is essential. If the distribution were integrable, then, at least locally, there would be a submanifold S of M tangent to D, in which case (S, g) becomes a Riemannian manifold and the theory D falls in the case of a very well known class of manifolds.

One may denote the sub-Riemannian manifold by the triplet (M, D, g).