Let $M$ be a closed, orientable 3manifold with a nontrivial differentiable $S^1$action.
What does this imply for $M$? What are examples except for (products of) spheres?
Let $M$ be a closed, orientable 3manifold with a nontrivial differentiable $S^1$action. What does this imply for $M$? What are examples except for (products of) spheres? 


You will find the complete answer in the papers and 


If the action has no fixed points, that class of manifolds has a name: Seifertfibred 3manifold with an orientation on the fiber. Seifert classified a slightly larger class of manifolds (Seifertfibered ones  no orientability constraint on the fibers). This is available in either the Jaco or Hatcher 3manifolds lecture notes, also Orlik's book on SeifertFibered spaces. Such examples include things like lens spaces, and relatively complicated 3manifolds like the double of a torus knot complement. When you allow fixed points, the orbit decomposition theorem gives you a stratification of the manifold into the fixedpoint set (a link) together with its Seifertfibred complement. So you could view these manifolds as certain Dehn Fillings on Seifertfibred manifolds with boundary. As the references above point out, they're abundant. 

