# Circle foliations not induced by circle actions on an compact orientable manifold

It is known that if we have an orientable fiber bundle $$E\to B$$, with fiber a circle $$\mathbb{S}^1$$, then it is a principal $$SO(2)$$-bundle. In other words, the fibers are spanned by the orbits of a circle action.

Now consider a regular foliation $$(E,\mathcal{F})$$ by circles, on $$E$$ an oriented compact manifold. By regular foliation I mean a partition into embedded submanifolds, each homeomorphic to a circle, and with tangent space spanned by a vector field.

They differ from fiber bundles in the following way: when we consider the leaf space $$E/\mathcal{F}$$ induced by the partition, it has the structure of an orbifold (see Theorem 2.5 in Introduction to foliations and Lie groupoids (Moerdijk & Marcun, Cambridge). Thus in some sense we have a circle bundle over an orbifold.

Question 1: Does an explicit example exists of a one of such foliations, that is not coming from a circle action?

The only examples of non homogeneous foliations by circles I am aware of, are on non-orientable manifolds, or non-compact ones.

• Yes. – Mike Miller Apr 15 at 14:17
• Nonorientable circle bundle over nonorientable surface base (so that the total space is orientable) is another example. More generally, orientable Seifert 3-manifold with non-orientable base. – Misha Apr 15 at 15:35