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?



Is every orientable circle bundle principal?
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.


