Chow–Rashevskii theorem
In sub-Riemannian geometry, the Chow–Rashevskii theorem asserts that any two points of a connected sub-Riemannian manifold are connected by a horizontal path in the manifold. It is named after Wei-Liang Chow who proved it in [|1939], and Petr Konstanovich Rashevskii, who proved it independently in [|1938].
The theorem has a number of equivalent statements, one of which is that the topology induced by the Carnot–Carathéodory metric is equivalent to the intrinsic topology of the manifold. A stronger statement that implies the theorem is the ball-box theorem. See, for instance, and.
