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 $$M$$ 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 $$\dot\gamma(t)\in H_{\gamma(t)}(M)$$ 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 $$A(x),\ [A,B](x),\ [A,[B,C]](x),\ [A,[B,[C,D]]](x),\dotsc\in T_x(M)$$ where all vector fields $$A,B,C,D, \dots$$ are horizontal. 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
 * $$d(x, y) = \inf\int_0^1 \sqrt{g(\dot\gamma(t),\dot\gamma(t))} \, dt,$$

where infimum is taken along all horizontal curves $$\gamma: [0, 1] \to M$$ such that $$\gamma(0)=x$$, $$\gamma(1)=y$$. 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
 * $$\mathbb R^2\times S^1.$$

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
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.