# Converse to Chow's theorem in sub-riemannian geometry

Chow's theorem is the statement that if $M$ is a connected smooth manifold endowed with a distribution $\mathcal{D}$ which is completely non integrable (i.e. iterated commutators of smooth sections of $\mathcal{D}$ generate the full tangent bundle $TM$), then any pair of points $x,y\in M$ may be connected by a piecewise smooth horizontal curve $\gamma$, i.e. a curve such that $\gamma'(t)\in \mathcal{D}_{\gamma(t)}$ for every $t$.

The converse statement (every pair of points can be connected by horizontal curves implies that the distribution is completely non integrable) is false in general and holds under an additional assumption of analyticity (of $M$ and $\mathcal{D}$). I need to use this fact in a paper and I realized I do not know of any place where one can find a proof (or probably I have a bad memory). Do you know of any citable source?

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Velimir Jurdjevic, Geometric Control Theory, p. 48, theorem 6: the orbits of any Lie algebra of real analytic vector fields have tangent space at each point given by the values of those vector fields at that point, so a more general result which immediately implies what you are looking for.

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Thank you very much! Unfortunately there are many "linguistic" differences between Control Theory texts and Subriemannian Geometry ones that may make not very appropriate to quote Jurdjevic book while discussing subriemannian geometry. Do you know of any "subriemannian" reference for the result I look for? Thank you very much! – Gian Maria Dall'Ara Jul 14 '13 at 8:57
Sorry, I can't think of one. I don't know the literature in either subject. – Ben McKay Jul 16 '13 at 12:52
Thank you anyway! – Gian Maria Dall'Ara Jul 17 '13 at 9:03

I think the correct reference should be Theorem 1 in

Tadashi NAGANO, Linear differential systems with singularities and an application to transitive Lie algebras, J. Math. Soc. Japan Volume 18, Number 4 (1966), 398-404.

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That is an earlier reference, but more difficult to read. It depends on whether the reference is intended to guide the reader to understand the proof or to give credit for discovery of the proof. – Ben McKay Jul 30 '13 at 19:32
that is true, I suggested it since it was needed for a citation in a paper and the paper is more geometrically oriented and less control theoretical, as it was requested. – Dario Prandi Jul 31 '13 at 8:19
Thank you, Dario! – Gian Maria Dall'Ara Aug 5 '13 at 15:57