Sub-Riemannian manifold

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

A distribution on $H (M)$ is called completely non-integrable if for any $x\in M$ we have that any tangent vector can be presented as a linear combination of vectors of the following types $A(x),\ [A,B](x),\ [A,[B,C]](x),\ [A,[B,[C,D]]](x),...\in T_x(M)$ 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

$d(x, y) = \inf\int_0^1 \sqrt{g(\dot\gamma(t),\dot\gamma(t))} ,$

where infimum is taken along all horizontal curves $\gamma: [0, 1] \to M$ such that $γ(0) = x$ , $γ(1) = y$ .

Categories: Metric geometry | Riemannian geometry

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Formal definition