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 Msuch 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 $$alpha and $$eta in the corresponding Lie algebra such that $${ alpha,eta, [alpha,eta] } spans the entire algebra. The horizontal distribution $$H spanned by left shifts of $$alpha and $$eta 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.