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 may be understood in the language of sub-Riemannian geometry. The Heisenberg group, important to quantum mechanics, carries a natural sub-Riemannian structure.

Definitions[edit]

By a distribution on we mean a subbundle of the tangent bundle of .

Given a distribution a vector field in is called horizontal. A curve on is called horizontal if for any .

A distribution on is called completely non-integrable if for any we have that any tangent vector can be presented as a linear combination of vectors of the following types where all vector fields are horizontal.

A sub-Riemannian manifold is a triple , where is a differentiable manifold, is a completely non-integrable "horizontal" distribution and is a smooth section of positive-definite quadratic forms on .

Any sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot–Carathéodory, defined as

where infimum is taken along all horizontal curves such that , .

Examples[edit]

A position of a car on the plane is determined by three parameters: two coordinates and for the location and an angle which describes the orientation of the car. Therefore, the position of the car can be described by a point in a manifold

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

A closely related example of a sub-Riemannian metric can be constructed on a Heisenberg group: Take two elements and in the corresponding Lie algebra such that

spans the entire algebra. The horizontal distribution spanned by left shifts of and is completely non-integrable. Then choosing any smooth positive quadratic form on gives a sub-Riemannian metric on the group.

Properties[edit]

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 is given by the Chow–Rashevskii theorem.

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