Sub-Riemannian manifold explained

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

M we mean a subbundle of the tangent bundle of M (see also distribution).

Given a distribution

H(M)\subset T(M) a vector field in H(M) is called horizontal. A curve \gamma on M is called horizontal if • \gamma(t)\in H _{\gamma(t)} (M) for any t .

A distribution on

H(M) is called completely non-integrable or bracket generating if for any 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 $$A(x),\backslash \; [A,B](x),\backslash \; [A,[B,C]](x),\backslash \; [A,[B,[C,D]]](x),\backslash dotsc\backslash in\; T\_x(M)$$ where all vector fields A,B,C,D, ... are horizontal. This requirement is also known as Hörmander's condition.

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

d(x, y) = inf \int ^{1} _{0} \sqrt{g( • • \gamma(t), \gamma(t))} dt, where infimum is taken along all horizontal curves \gamma: [ 0 , 1 ] \to M such that \gamma( 0 ) = x , \gamma( 1 ) = y .Horizontal curves can be taken either Lipschitz continuous, Absolutely continuous or in the Sobolev space H ^{1([0,1],M)} producing the same metric in all cases. The fact that the distance of two points is always finite (i.e. any two points are connected by an 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

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 R ^{2 x} 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 x} 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

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.

See also

• Carnot group, a class of Lie groups that form sub-Riemannian manifolds.
• Distribution
• Hörmander's condition
• Optimal control