Quantum

In the Riemannian world all directions in space are available to you. One can move along an arbitrary curve or calculate the scalar product between any two vectors. This makes Riemannian manifolds similar to Euclidean spaces, and indeed, Euclidean spaces provide excellent local approximations.

In a sub-Riemannian manifold it is possible to measure distances only along certain directions, which forces us to consider only special curves, known as the horizontal curves, whose tangent vectors have finite length. As a result, sub-Riemannian manifolds exhibit a variety of different phenomena not present in the Riemannian case.