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, 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 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).
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), (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, ISBN978-0821841655
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
chow, rashevskii, theorem, riemannian, geometry, also, known, chow, theorem, asserts, that, points, connected, riemannian, manifold, endowed, with, bracket, generating, distribution, connected, horizontal, path, manifold, named, after, liang, chow, proved, 193. 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 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 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 Caratheodory 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 editOrbit control theory References editChow W L 1939 Uber Systeme von linearen partiellen Differentialgleichungen erster Ordnung Mathematische Annalen 117 98 105 doi 10 1007 bf01450011 S2CID 121523670 Gromov M 1996 Carnot Caratheodory spaces seen from within PDF in A Bellaiche ed Proc Journees nonholonomes geometrie sous riemannienne theorie du controle robotique Paris France June 30 July 1 1992 Prog Math vol 144 Birkhauser 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 nbsp This differential geometry related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Chow Rashevskii theorem amp oldid 1200516869, wikipedia, wiki, book, books, library,
