Orbits of families of vector fields and integrability of distributions

Abstract:

Let $D$ be an arbitrary set of ${C^\infty }$ vector fields on the ${C^\infty }$ manifold $M$. It is shown that the orbits of $D$ are ${C^\infty }$ submanifolds of $M$, and that, moreover, they are the maximal integral submanifolds of a certain ${C^\infty }$ distribution ${P_D}$. (In general, the dimension of ${P_D}(m)$ will not be the same for all $m \in M$.) The second main result gives necessary and sufficient conditions for a distribution to be integrable. These two results imply as easy corollaries the theorem of Chow about the points attainable by broken integral curves of a family of vector fields, and all the known results about integrability of distributions (i.e. the classical theorem of Frobenius for the case of constant dimension and the more recent work of Hermann, Nagano, Lobry and Matsuda). Hermann and Lobry studied orbits in connection with their work on the accessibility problem in control theory. Their method was to apply Chow’s theorem to the maximal integral submanifolds of the smallest distribution $\Delta$ such that every vector field $X$ in the Lie algebra generated by $D$ belongs to $\Delta$ (i.e. $X(m) \in \Delta (m)$ for every $m \in M$). Their work therefore requires the additional assumption that $\Delta$ be integrable. Here the opposite approach is taken. The orbits are studied directly, and the integrability of $\Delta$ is not assumed in proving the first main result. It turns out that $\Delta$ is integrable if and only if $\Delta = {P_D}$, and this fact makes it possible to derive a characterization of integrability and Chow’s theorem. Therefore, the approach presented here generalizes and unifies the work of the authors quoted above.