Typical Properties of Leaves of Cartan Foliations with Ehresmann Connection

Abstract

We consider a Cartan foliation (M,F) of an arbitrary codimension q admitting an Ehresmann connection such that all leaves of (M,F) are embedded submanifolds of M. We prove that for any foliation (M,F) there exists an open, not necessarily connected, saturated, and everywhere dense subset M₀ of M and a manifold L₀ such that the induced foliation (M₀, F_M0) is formed by the fibers of a locally trivial fibration with the standard fiber L₀ over (possibly, non-Hausdorff) smooth q-dimensional manifold. In the case of codimension 1, the induced foliation on each connected component of the manifold M₀ is formed by the fibers of a locally trivial fibration over a circle or over a line.