Sub-Riemannian manifold

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).

Sub-Riemannian manifolds often occur in the study of constrained systems in classical mechanics, such as the motion of vehicles on a surface, the motion of robot arms, and the orbital dynamics of satellites. Geometric quantities such as the Berry phase may be understood in the language of sub-Riemannian geometry. The Heisenberg group, important to quantum mechanics, carries a natural sub-Riemannian structure.

Definitions

By a distribution on ${\displaystyle M}$ we mean a subbundle of the tangent bundle of ${\displaystyle M}$ (see also distribution).

Given a distribution ${\displaystyle H(M)\subset T(M)}$ a vector field in ${\displaystyle H(M)}$ is called horizontal. A curve ${\displaystyle \gamma }$ on ${\displaystyle M}$ is called horizontal if ${\displaystyle \dot{\gamma }(t)\in H_{\gamma (t)}(M)}$ for any ${\displaystyle t}$.

A distribution on ${\displaystyle H(M)}$ is called completely non-integrable or bracket generating if for any ${\displaystyle x\in M}$ we have that any tangent vector can be presented as a linear combination of Lie brackets of horizontal fields, i.e. vectors of the form $${\displaystyle A(x),[A,B](x),[A,[B,C]](x),[A,[B,[C,D]]](x),\dots \in T_x(M)}$$ where all vector fields ${\displaystyle A,B,C,D,\dots }$ are horizontal. This requirement is also known as Hörmander's condition.

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

Any (connected) sub-Riemannian manifold carries a natural intrinsic metric, called the metric of Carnot–Carathéodory, defined as

${\displaystyle d(x,y)=inf{\int }_0^1\sqrt{g(\dot{\gamma }(t),\dot{\gamma }(t))}dt,}$

where infimum is taken along all horizontal curves ${\displaystyle \gamma :[0,1]\to M}$ such that ${\displaystyle \gamma (0)=x}$, ${\displaystyle \gamma (1)=y}$. Horizontal curves can be taken either Lipschitz continuous, Absolutely continuous or in the Sobolev space ${\displaystyle H^1([0,1],M)}$ 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.