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

Let H, K be a couple of vector fields of class C1 in an open set U ⊂ ℝN+m, ℳ be a N-dimensional C1 submanifold of U and define T: = { z∈ ℳ: H(z) , K(z) ∈ Tzℳ}. Then the obvious property If z0 ∈ ℳ is an interior point (relative to ℳ)of T then [H,K](z0)∈Tz0ℳ admits the following generalization: If z0 ∈ ℳ is a superdensity point (relative to ℳ)of T then [H,K](z0)∈Tz0ℳ. As a corollary we get very easily the following result of [7]: Let D be a C1 distribution of rank N on an open set U ⊂ ℝN+m and let ℳ be a N-dimensional C1 submanifold of U. If z0 ∈ ℳ is a superdensity point (relative to ℳ)of the tangency set { z∈ ℳ: Tzℳ= D(z) } then D is involutive at z0.