The neighborhood of a singular leaf (Q2034723)

This paper is concerned with the local description of a manifold endowed with a singular foliation, about a singular leaf \(L\). Near a simply connected regular leaf, it is well known that holonomy forces the manifold to be formally semi-locally trivial. For a singular leaf, the situation is very different. On one hand, the holonomy group of a singular leaf is a non-trivial Lie group. On the other hand, there is a transverse foliation which is singular as well. The author first shows that when \(L\) is simply connected, triviality about \(L\) follows if the transverse foliation comprises vector fields which vanish at second order, at least. When the leaf \(L\) is 2-connected and the transverse linear part is not trivial, the author still obtains a local triviality result in the form of a Levi-Malcev theorem for the semi-simple part of the holonomy Lie algebra.