Classification of free $S^1$-actions on $S^2\times S^1$

Question

Consider smooth and free actions of the circle $S^1$ on the manifold $S^2\times S^1$. Are there any known classification of these up to equivariant diffeomorphism? One example of such an action is by simply letting the $S^1$ act in the obvious way on the $S^1$ factor, and leaving the $S^2$ factor intact. Could it possibly be the case that this is the only action (up to equivariant diffeomorphism)?

Answer 1

Only one. One way to see this is to note that such an action on a manifold $M$ amounts to giving $M$ structure of a principal $S^1$-bundle. It is easy to see that in the case $M=S^2\times S^1$ the base has to be $S^2$. Now, use the classification of $S^1$ bundles over $S^2$ in terms of their Euler classes to conclude that the bundle has to be trivial.

You can find this type of arguments (and more) in any book dealing with "Seifert manifolds" (say, Orlik's). One can prove even more, that there is only one circle foliation on $S^2\times S^1$, given by a free circle action, up to isotopy.