Sub-Riemannian manifold
From Free net encyclopedia
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).
Contents |
Definitions
By a distribution on <math>M</math> we mean a subbundle of the tangent bundle of <math>M</math>.
Given a distribution <math>H(M)\subset T(M)</math> a vector field in <math>H(M)\subset T(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 if for any <math>x\in M</math> we have that any tangent vector can be presented as a linear combination of vectors of the following types <math>A(x),\ [A,B](x),\ [A,[B,C]](x),\ [A,[B,[C,D]]](x),...\in T_x(M)</math> where all vector fields <math>A,B,C,D, ...</math> are horizontal.
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 forms on <math>H</math>.
Any sub-Riemannian manifold carries the 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))}
,</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>.
Examples
A position of a car on the plane is determined by three parameters: two coordinates <math>x</math> and <math>y</math> for the location and an angle <math>\alpha</math> which describes the orientation of the car. Therefore, the position of car can be described by a point in a manifold <math>\mathbb R^2\times S^1</math>. 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 <math>\mathbb R^2\times S^1</math>.
Closely related example of sub-Riemannian metric can be constructed on Heisenberg group: Take two elements in corresponding Lie algebra <math>\alpha,\beta</math>, such that <math>\alpha,\beta,[\alpha,\beta]</math> span all algebra. Then horizontal distribution <math>H</math> spanned by left shifts of <math>\alpha</math> and <math>\beta</math> is completely non-integrable. Then one has to choose any smooth positive quadratic form on <math>H</math>.
Properties
For every sub-Riemannian manifold, there exists a Hamiltonian, called the sub-Riemannian Hamiltonian, constructed out of the cometric 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.