fiber bundles with both base and fiber as $S^1$.

Question

What are all fiber bundles with both base and fiber as $S^1$ ? I know torus and Klein bottle. I think there are only two of them. But I dont know how to prove. How to classify them ? I am searching for some easy argument to prove such.

Answer 1

Such a bundle is a 2-dimensional closed manifold with a 1-dimensional foliation which is a codimension 1 here. A theorem of Thurston implies that the euler characteristic of such a manifold is zero. So it is the torus or the Klein bottle.

http://www.jstor.org/stable/1971047?seq=1#page_scan_tab_contents

Answer 2

Here is an alternative answer without foliations : for any fiber bundle $f : E \to B$ with fiber $F$ we have $\chi(E) = \chi(F) \chi(B)$. In your particular, $\chi(E) = 0$ so $E$ can only be the Klein bottle or the torus by the classification of compact surfaces.