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 ∈ 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 Cᵏ, 2 k ∞, or k = ω, there exists a completely controllable set S consisting of two vector fields of class Ck - 1.