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Chow–Rashevskii theorem
In sub-Riemannian geometry, the Chow–Rashevskii theorem (also known as Chow's 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 (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).
References
Chow, W.L. (1939), "Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung", Mathematische Annalen 117: 98–105, doi:10.1007/bf01450011
Gromov, M. (1996), "Carnot-Carathéodory spaces seen from within", in A. Bellaiche, Proc. Journées nonholonomes: géométrie sous-riemannienne, théorie du contrôle, robotique, Paris, France, June 30--July 1, 1992. (PDF), Prog. Math. 144, Birkhäuser, Basel, pp. 79–323
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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