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-Caratheodory. The Hausdorff dimension of such metric spaces is always an integer and larger than its topological dimension (unless it is actually a Riemannian manifold).

Formal definition

By a distribution on M we mean a subbundle of the tangent bundle of M.

Given a distribution a vector field in is called horizontal. A curve γ on M is called horizontal if for any t.

A distribution on H(M) is called completely non-integrable if for any we have that any tangent vector can be presented as a linear combination of vectors of the following types where all vector fields A,B,C,D,... are horizontal.

A sub-Riemannian manifold is a triple (M,H,g), where M, is a differentiable manifold, H is a completely non-integrable "horizontal" distribution and g is a section of positive-definite quadratic forms on H.

Any sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot-Caratheodory, defined as

where infimum is taken along all horizontal curves such that γ(0) = x, γ(1) = y.