TITLE: Is every orientable circle bundle principal?
QUESTION [10 upvotes]: The only examples I found of nonprincipal circle bundle are nonorientable, like the Klein bottle that is an S^1 bundle over S^1 which is not principal and nontrivial. That makes me ask the question.
Is it true that every orientable circle bundle is principal?

REPLY [17 votes]: Yes,it is true.You can find this result from "Geometry of differential forms" by Morita (Page 241)
PROPOSITION 6.15. Every oriented $S^1$ bundle admits the structure of principal $S^1$ bundle.