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 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_{\gamma(t)}(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),\dotsc\in T_x(M) where all vector fields A,B,C,D, \dots 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{g(\dot\gamma(t),\dot\gamma(t))} \, dt,

where infimum is taken along all horizontal curves \gamma: [0, 1] \to M such that \gamma(0)=x, \gamma(1)=y.

Examples[edit]

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 the 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.

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

\{ \alpha,\beta,[\alpha,\beta]\}

spans the entire algebra. The horizontal distribution H spanned by left shifts of \alpha and \beta is completely non-integrable. Then choosing any smooth positive quadratic form on H 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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