In sub-Riemannian geometry, the Chow–Rashevsky theorem (also known as Chow's theorem) asserts that any two points of a connected sub-Riemannian manifold, endowed with a bracket generating distribution, are connected by a horizontal path in the manifold. It is named after Wei-Liang Chow who proved it in 1939, and Pyotr Rashevsky, 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 (locally Euclidean) topology of the manifold. A stronger statement that implies the theorem is the ball–box theorem. See, for instance, Montgomery (2006) and Gromov (1996).
See also Orbit (control theory)
References Chow, W.L. (1939), "Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung", Mathematische Annalen, 117: 98–105, doi:10.1007/bf01450011, S2CID 121523670 Gromov, M. (1996), "Carnot-Carathéodory spaces seen from within" (PDF), in A. Bellaiche (ed.), Proc. Journées nonholonomes: géométrie sous-riemannienne, théorie du contrôle, robotique, Paris, France, June 30--July 1, 1992., Prog. Math., vol. 144, Birkhäuser, Basel, pp. 79–323, archived from the original (PDF) on September 27, 2011, retrieved January 27, 2013 Montgomery, R. (2006), A tour of sub-Riemannian geometries: their geodesics and applications, American Mathematical Society, ISBN 978-0821841655 Rashevskii, P.K. (1938), "About connecting two points of complete non-holonomic space by admissible curve (in Russian)", Uch. Zapiski Ped. Inst. Libknexta (2): 83–94