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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.

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    $\begingroup$ Yes. $\endgroup$ – Mike Miller Apr 15 at 14:17
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    $\begingroup$ 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. $\endgroup$ – Misha Apr 15 at 15:35

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