Involutivity degree of a distribution at superdensity points of its tangencies

Let {equation presented} be vector fields of class Ck in an open set {equation presented} be a N-dimensional Ck submanifold of U and define {equation presented} where TzM is the tangent space to M at z. Then we expect the following property, which is obvious in the special case when z0 is an interior point (relative to M) of T: If z0 ϵ M is a (N + k)-density point (relative to M) of T then all the iterated Lie brackets of order less or equal to k {equation presented} (h, ih ≤ k+1) belong to Tz0M. Such a property has been proved in [9] for k = 1 and its proof in the case k = 2 is the main purpose of the present paper. The following corollary follows at once: Let D be a C2 distribution of rank N on an open set U ⊂ RN+m and M be a N-dimensional C2 submanifold of U. Moreover let z0 ϵ M be a (N + 2)-density point of the tangency set {z ϵ M| TzM = D(z)}. Then D must be 2-involutive at z0, i.e., for every family {Xj}Nj =1 of class C2 in a neighborhood V ⊂ U of z0 which generates D one has {equation presented}. © Masaryk University, 2021.