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–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, are best 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 $M$ 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 $gamma$ on $M$ is called horizontal if $dotgamma(t)in\; H\_\{gamma(t)\}(M)$ for any $t$.

A distribution on $H(M)$ is called completely non-integrableif for any $xin\; M$ 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 smooth section of positive-definite quadratic forms on $H$.

Any sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot–Carathéodory, defined as ::$d(x,\; y)\; =\; infint\_0^1\; sqrt\{g(dotgamma(t),dotgamma(t))\},$where infimum is taken along all "horizontal curves" $gamma:\; [0,\; 1]\; o\; M$such that $gamma(0)=x$, $gamma(1)=y$.

Examples

A position of a car on the plane is determined by three parameters: two coordinates $x$ and $y$ for the location and an angle $alpha$ which describes the orientation of the car.Therefore, the position of car can be described by a point in a manifold $mathbb\; R^2\; imes\; S^1$.One can ask what is the minimal distance one should drive to get from one position to another; this defines a Carnot–Carathéodory metric on the manifold $mathbb\; R^2\; imes\; S^1$.

Closely related example of sub-Riemannian metric can be constructed on a Heisenberg group: Take two elements $alpha$ and in the corresponding Lie algebra such that spans the entire algebra. The horizontal distribution $H$ spanned by left shifts of $alpha$ and is "completely non-integrable".Then choosing any smooth positive quadratic form on $H$ gives a sub-Riemannian metric on the group.

Properties

For every sub-Riemannian manifold, there exists a Hamiltonian, called the sub-Riemannian Hamiltonian, constructed out of the metric for the manifold. Conversely, every such quadratic Hamiltonian induces a sub-Riemannian manifold. The existence of geodesics of the corresponding Hamilton–Jacobi equations for the sub-Riemannian Hamiltonian are given by the Chow–Rashevskii theorem.

References

* Richard Montgomery, "A Tour of Subriemannian Geometries, Their Geodesics and Applications (Mathematical Surveys and Monographs, Volume 91)", (2002) American Mathematical Society, ISBN 0-8218-1391-9.