Chow–Rashevskii theorem

Short description: On horizontal paths in a sub-Riemannian manifold

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).

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 
• 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., Prog. Math., 144, Birkhäuser, Basel, pp. 79–323, https://www.ihes.fr/~gromov/PDF/carnot_caratheodory.pdf, 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