The question I would like to answer is the following.
From the classification of sphere bundles we know that the only orientable $S^3$ bundle over $S^1$ is $S^3 \times S^1$.
So suppose we have lens space bundle $L_n(1) \rightarrow P \rightarrow S^1$, where $L_n(1)=S^3/\mathbb{Z}_n$ is the lens space sometimes also denoted by $L(n;1)$, and $P$ is oreintable.
Does it follow from bundle theory that $P=L_n(1) \times S^1$?
My nose tells me this is true, but I know too little about bundle theory to answer it on my own.