Are Seifert fibered spaces with a horizontal surface exactly the surface bundles over a circle with periodic monodromy? 


I am unsure of my arguments for this: If a SFS has a horizontal surface then splitting along this surface gives a $(surface \times I)$ with a periodic monodromy. Conversely given a surface bundle over a circle with periodic monodromy the flow lines of the monodromy give a foliation of the manifold with circles, with the surface fiber as a horizontal surface.



Are Seifert fibered spaces with a horizontal surface exactly the surface bundles over the circle with periodic monodromy?
This is not quite true, because after splitting along the horizontal surface you may get interval bundles over non-orientable surfaces, and these are not products (I am assuming your 3-manifold is orientable). This holds for instance if the manifold is a circle bundle over a non-orientable surface (with Euler number zero, so that it has indeed a horizontal surface).


If I am not missing something, the following facts should be equivalent on an orientable Seifert fibered space $M$ without boundary:


1. The fibration has a horizontal surface
- The fibration has Euler number zero
- The manifold $M$ either fibers over $S^1$ with periodic monodromy, or it fibers over the interval orbifold, in such a way that its natural double cover fibers over $S^1$ with periodic monodromy
- The manifold $M$ has some cover (of any finite degree) that fibers over $S^1$ with periodic monodromy
- The manifold $M$ is finitely covered by $S\times S^1$ for some closed orientable surface $S$.
- The manifold $M$ has a geometry of type $S^2 \times \mathbb R$, $\mathbb R^3$, or $\mathbb H^2 \times \mathbb R$


There are few cases where the fibration is not unique, but "having a horizontal surface" or "having Euler number zero" are properties that hold for some fibration if and only if they hold for any fibration.


