Foliations with few non-compact leaves

Abstract

Let $\left(F\right)$ be a foliation of codimension 2 on a compact manifold with at least one non-compact leaf. We show that then $\left(F\right)$ must contain uncountably many non-compact leaves. We prove the same statement for oriented $p$–dimensional foliations of arbitrary codimension if there exists a closed $p$ form which evaluates positively on every compact leaf. For foliations of codimension 1 on compact manifolds it is known that the union of all non-compact leaves is an open set [A Haefliger, Varietes feuilletes, Ann. Scuola Norm. Sup. Pisa 16 (1962) 367–397].