any two points on contact manifold connected by a path tangent to the contact distribution

Question

Let $(M, B)$ be a manifold with a subbundle $B \subset TM$ such that the Frobenius tensor $B \times B \to TM/B$ given by the commutator of vector fields is non-degenerate. Is it true that any two points $x, y \in M$ can be connected by a piecewise smooth path tangent to $B$?

Answer 1

The answer yes, by the Chow–Rashevskii theorem. (That theorem applies more generally, to sub-Riemannian metrics whose horizontal distributions are bracket-generating, meaning that every tangent vector at every point can be expressed as a Lie bracket of finitely many vector fields tangent to the horizontal distribution. Contact manifolds always satisfy this condition.)