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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 <math>M<math> we mean a subbundle of the tangent bundle of <math>M<math>.
Given a distribution <math>H(M)\subset T(M)<math> a vector field
in <math>H(M)\subset T(M)<math> is called horizontal. A curve <math>\gamma<math> on <math>M<math> is called horizontal if <math>\dot\gamma(t)\in H_{\gamma(t)}(M)<math> for any
<math>t<math>.
A distribution on <math>H(M)<math> is called completely non-integrable
if for any <math>x\in M<math> we have that any tangent vector canbe presented as a linear combination of vectors of the following types
<math>A(x),\ [A,B](x),\ [A,[B,C]](x),\ [A,[B,[C,D]]](x),...\in T_x(M)<math> where all vector fields <math>A,B,C,D, ...<math> are horizontal.
A sub-Riemannian manifold is a triple <math>(M, H, g)<math>,
where <math>M,<math> is a differentiable manifold,
<math>H<math> is a completely non-integrable "horizontal" distribution
and <math>g<math> is a section of positive-definite quadratic forms on <math>H<math>.
Any sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot-Caratheodory, defined as
- <math>d(x, y) = \inf\int_0^1 \sqrt{g(\dot\gamma(t),\dot\gamma(t))}
,<math>
where infimum is taken along all horisontal curves <math>\gamma: [0, 1] \to M<math>
such that <math>\gamma(0)=x<math>, <math>\gamma(1)=y<math>.
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