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Type of generalization of a Riemannian manifold

Sub-Riemannian manifold

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

By a distribution on {\displaystyle M} we mean a subbundle of the tangent bundle of {\displaystyle M} (see also distribution).

Given a distribution {\displaystyle H(M)\subseteq T(M)} a vector field in {\displaystyle H(M)} is called horizontal. A curve {\displaystyle \gamma } on {\displaystyle M} is called horizontal if {\displaystyle {\dot {\gamma }}(t)\in H_{\gamma (t)}(M)} for any {\displaystyle t}.

A distribution {\displaystyle H(M)} is called completely non-integrable or bracket generating if for any {\displaystyle x\in M} we have that any tangent vector can be presented as a linear combination of Lie brackets of horizontal fields, i.e. vectors of the form {\displaystyle A(x),\ [A,B](x),\ [A,[B,C]](x),\ [A,[B,[C,D]]](x),\dotsc \in T_{x}(M)} where all vector fields {\displaystyle A,B,C,D,\dots } are horizontal. This requirement is also known as Hörmander's condition.

A sub-Riemannian manifold is a triple {\displaystyle (M,H,g)}, where {\displaystyle M} is a differentiable manifold, {\displaystyle H} is a completely non-integrable "horizontal" distribution and {\displaystyle g} is a smooth section of positive-definite quadratic forms on {\displaystyle H}.

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

{\displaystyle d(x,y)=\inf \int _{0}^{1}{\sqrt {g({\dot {\gamma }}(t),{\dot {\gamma }}(t))}}\,dt,}

where infimum is taken along all horizontal curves {\displaystyle \gamma:[0,1]\to M} such that {\displaystyle \gamma (0)=x}, {\displaystyle \gamma (1)=y}. Horizontal curves can be taken either Lipschitz continuous, absolutely continuous or in the Sobolev space {\displaystyle H^{1}([0,1],M)} producing the same metric in all cases.

The fact that the distance between two points is always finite (i.e. any two points are connected by a horizontal curve) is a consequence of Hörmander's condition known as Chow–Rashevskii theorem.

Examples

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

{\displaystyle \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

{\displaystyle \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 {\displaystyle \alpha } and {\displaystyle \beta } in the corresponding Lie algebra such that

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

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

Solutions of the corresponding Hamilton–Jacobi equations for the sub-Riemannian Hamiltonian are called geodesics, and generalize Riemannian geodesics.

Read next

Finsler manifoldGeneralization of Riemannian manifolds Riemann curvature tensorTensor field in Riemannian geometry Parallel transportSystem of moving vectors in differential geometry Spray (mathematics)Vector field on tangent bundle Biharmonic map Geodesics as Hamiltonian flows

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