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, {{harvtxt|Montgomery|2006}} and {{harvtxt|Gromov|1996}}.
See also
References
- {{citation|first=W.L.|last=Chow|title=Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung|year=1939|journal=Mathematische Annalen|volume=117|pages=98–105|doi=10.1007/bf01450011 |s2cid=121523670}}
- {{citation|first=M.|last=Gromov|authorlink=Mikhail Leonidovich Gromov|chapter-url=https://www.ihes.fr/~gromov/PDF/carnot_caratheodory.pdf|title=Proc. Journées nonholonomes: géométrie sous-riemannienne, théorie du contrôle, robotique, Paris, France, June 30--July 1, 1992.|editor=A. Bellaiche|series=Prog. Math.|volume=144|year=1996|pages=79–323|publisher=Birkhäuser, Basel|chapter=Carnot-Carathéodory spaces seen from within|access-date=January 27, 2013|archive-url=https://web.archive.org/web/20110927014016/http://www.ihes.fr/~gromov/PDF/carnot_caratheodory.pdf|archive-date=September 27, 2011|url-status=dead}}
- {{citation|first=R.|last=Montgomery|title=A tour of sub-Riemannian geometries: their geodesics and applications|publisher=American Mathematical Society|year=2006|isbn=978-0821841655}}
- {{citation|first=P.K.|last=Rashevskii|year=1938|title=About connecting two points of complete non-holonomic space by admissible curve (in Russian)|journal=Uch. Zapiski Ped. Inst. Libknexta|issue=2|pages=83–94}}
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