ChowRashevskii

ChowRashevskii, often referred to as the Chow–Rashevskii theorem, is a fundamental result in sub-Riemannian geometry and geometric control theory. It concerns manifolds equipped with a distribution and a metric defined only on that distribution.

A sub-Riemannian manifold consists of a smooth manifold M, a smooth distribution Δ, which is a subbundle of the tangent bundle TM, and an inner product on each fiber Δp. A curve γ in M is called horizontal if its tangent vector γ′(t) lies in Δγ(t) for almost every t. The length of a horizontal curve is computed using the metric on Δ, and the distance between two points is defined as the infimum of the lengths of horizontal curves joining them; this distance is known as the Carnot–Carathéodory distance.

The Lie bracket-generating (or Hörmander) condition requires that the Lie algebra generated by smooth sections of Δ span the entire tangent space at every point. Informally, by taking successive Lie brackets of vector fields in Δ, one can produce directions needed to reach any tangent direction.

Chow–Rashevskii theorem states that if M is connected and Δ is bracket-generating everywhere, then any two points in M can be connected by a horizontal curve. Consequently, the sub-Riemannian distance is finite for any pair of points and induces the manifold’s topology.

The theorem underpins controllability results in nonlinear control systems and justifies using horizontal curves to study the geometry of sub-Riemannian spaces. It is named after Wen-Chiao Chow and Russian mathematician Pavel Rashevsky, with origins in work from the 1930s–1950s.