Q:

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? 

A:

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.

