Sub-Riemannian manifold
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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, is one of the simplest examples of a sub-Riemannian manifold.
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 [\dot\gamma(t)\in H_(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 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) = \inf\int_0^1 \sqrt,]
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\times 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\times S^1].
Closely related example of sub-Riemannian metric can be constructed on Heisenberg group: Take two elements in corresponding Lie algebra [\alpha,\beta], such that [\alpha,\beta,[alpha,beta]] span all algebra. Then horizontal distribution [H] spanned by left shifts of [\alpha] and [\beta] is completely non-integrable. Then one has to choose any smooth positive quadratic form on [H].
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.
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