Editing Sub-Riemannian manifold (section)

==Definitions==

By a ''distribution'' on <math>M</math> we mean a [[subbundle]] of the [[tangent bundle]] of <math>M</math> (see also [[Distribution (differential geometry)|distribution]]).

Given a distribution <math>H(M)\subset T(M)</math> a vector field in <math>H(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'' or ''bracket generating'' if for any <math>x\in M</math> we have that any tangent vector can be presented as a [[linear combination]] of [[Lie bracket of vector fields|Lie brackets]] of horizontal fields, i.e. vectors of the form <math display="block">A(x),\ [A,B](x),\ [A,[B,C]](x),\ [A,[B,[C,D]]](x),\dotsc\in T_x(M)</math> where all vector fields <math>A,B,C,D, \dots</math> are horizontal. This requirement is also known as [[Hörmander's condition]].

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 smooth section of positive-definite [[quadratic form]]s on <math>H</math>.

Any (connected) sub-Riemannian manifold carries a natural [[intrinsic metric]], called the metric of Carnot–Carathéodory, defined as 
:<math>d(x, y) = \inf\int_0^1 \sqrt{g(\dot\gamma(t),\dot\gamma(t))} \, dt,</math>
where infimum is taken along all ''horizontal curves'' <math>\gamma: [0, 1] \to M</math> such that <math>\gamma(0)=x</math>, <math>\gamma(1)=y</math>.
Horizontal curves can be taken either [[Lipschitz continuous]], [[Absolutely continuous]] or in the [[Sobolev space]] <math> H^1([0,1],M) </math> producing the same metric in all cases.
 
The fact that the distance of two points is always finite (i.e. any two points are connected by an horizontal curve) is a consequence of Hörmander's condition known as [[Chow–Rashevskii theorem]].