Compact and locally dense leaves of a closed one-form foliation

Abstract

We study a foliation defined by a possibly singular smooth closed one-form on a connected smooth closed orientable manifold. We prove two bounds on the total number of homologically independent compact leaves and of connected components of the union of all locally dense leaves, which we call minimal components. In particular, we generalize the notion of minimal components, previously used in the context of Morse form foliations, to general foliations. Finally, we give a condition for the form foliation to have only closed leaves (closed in the complement of the singular set).