In
mathematics, a
sub-Riemannian manifold is a certain type of generalization of a
Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called
horizontal subspaces. Sub-Riemannian manifolds (and so,
a fortiori, Riemannian manifolds) carry a natural
intrinsic metric called the
metric of Carnot–Carathéodory. The
Hausdorff dimension of such
metric spaces is always an
integer and larger than its
topological dimension (unless it is actually a Riemannian manifold).