Sub-Riemannian manifold

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 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 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==