Good Behaviour of Lie Bracket at a Superdensity Point of the Tangency Set of a Submanifold With Respect to a Rank 2 Distribution

Abstract

Let H, K be a couple of vector fields of class C1 in an open set U ⊂ ℝN+m, $${\cal M}$$ be a N-dimensional C1 submanifold of U and define $${\cal T}: = \left\{{z \in {\cal M}:H\left(z \right),K\left(z \right) \in {T_z}{\cal M}} \right\}$$ . Then the obvious property If z0 ∈ $${\cal M}$$ is an interior point (relative to $${\cal M}$$ )of $${\cal T}$$ then $$\left[{H,K} \right]\left({{z_0}} \right) \in {T_{{z_0}}}{\cal M}$$ admits the following generalization: If z0 ∈ $${\cal M}$$ is a superdensity point (relative to $${\cal M}$$ )of $${\cal T}$$ then $$\left[{H,K} \right]\left({{z_0}} \right) \in {T_{{z_0}}}{\cal M}$$ . As a corollary we get very easily the following result of [7]: Let $${\cal D}$$ be a C1 distribution of rank N on an open set U ⊂ ℝN+m and let $${\cal M}$$ be a N-dimensional C1 submanifold of U. If z0 ∈ $${\cal M}$$ is a superdensity point (relative to $${\cal M}$$ )of the tangency set $$\left\{{z \in {\cal M}:{T_z}{\cal M} = {\cal D}\left(z \right)} \right\}$$ then $${\cal D}$$ is involutive at z0.