# Foliation with a compact leaf

Let $$M$$ be a closed oriented manifold, and $$F$$ be a fixed foliation of $$M$$. We assume the dimension and codimension of $$F$$ are both greater than $$1$$.

Q Under what condition, we can say that $$F$$ admits one compact leaf? Is there any research or result about this question ?

There are some results concerning your question that go in the negative direction. Namely, in the following paper two results are proven:

Counterexamples to the Seifert Conjecture and Opening Closed Leaves of Foliations Paul A. Schweitzer. Annals of Mathematics Second Series, Vol. 100, No. 2 (Sep., 1974), pp. 386-400. https://www.jstor.org/stable/1971077?seq=1#page_scan_tab_contents

Theorem D. If there exists any $$C^r$$ foliation $$F_0$$ of $$M$$ of codimension $$q\ge 3$$ with $$0\le r \le C^{\infty}$$, then there exists such a foliation $$F_1$$ with no closed leaves.

Theorem C. If there exists any $$C^r$$ foliation $$F_0$$ of $$M$$ of codimension two with $$r=0$$ or $$r=1$$, then there exists such a foliation $$F_1$$ with no closed leaves.

Moreover, $$F_0$$ and $$F_1$$ are related by a homothopy which is defined in the article of Schweitzer.

There is also a review of Lawson from about the same time.

Lawson. Foliations, Bulletin of the AMS Volume 80, Number 3, May 1974