On singular foliations tangent to a given hypersurface

We consider a class of singular foliations in the sense of Androulidakis and Skandalis that we call transverse order k foliations. These have a finite number of leaves: one hypersurface (the singular leaf) together with the components of its complement (open leaves). The positive integer parameter k encodes the “order of tangency” of the leafwise vector fields to L . We show that a loop in the singular leaf induces a well-defined holonomy transformation at the level of (k-1) -jets. The resulting holonomy invariant can be used to give a complete classification of these foliations and obtain concrete descriptions of their associated groupoids and algebras.