On Controllability by Means of Two Vector Fields

A set S of vector fields on a differentiable manifold M is said to be completely controllable if for every pair $(m,m')$ of points of M there exists a trajectory of S from m to $m'$. Here a trajectory of S is a curve which is an integral curve of some $X \in S$ or a finite concatenation of such curves so that, in general, a trajectory of S run in reverse is no longer a trajectory. Our main theorem is: on every connected paracompact manifold of class $C^k $, $2 \leqq k \leqq \infty $, or $k = \omega $, there exists a completely controllable set S consisting of two vector fields of class $C^{k - 1} $.