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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.

Contents

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

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

Given a distribution H ( M ) ⊂ T ( M ) a vector field in H ( M ) is called horizontal. A curve γ on M is called horizontal if γ ˙ ( t ) ∈ H γ ( t ) ( M ) for any t .

A distribution on H ( M ) is called completely non-integrable if for any x ∈ 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 ) , … ∈ 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 ∫ 0 1 g ( γ ˙ ( t ) , γ ˙ ( t ) ) d t ,

where infimum is taken along all horizontal curves γ : [ 0 , 1 ] → M such that γ ( 0 ) = x , γ ( 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 α which describes the orientation of the car. Therefore, the position of the car can be described by a point in a manifold

R 2 × 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

R 2 × S 1 .

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

{ α , β , [ α , β ] }

spans the entire algebra. The horizontal distribution H spanned by left shifts of α and β 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 is given by the Chow–Rashevskii theorem.

References

Sub-Riemannian manifold Wikipedia
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