Is there any example of a Riemannian submersion, which is no fibration?


As far as I know, a (any) submersion is locally, but not globally, given by a fibration. The converse holds globally. Nevertheless, I could not think of or find an example.



Examples of Riemannian Submersions
If $E\to B$ is a Riemannian submersion, then so is its restriction onto any open subset, which gives lots of examples of submersions that are not fibrations, e.g. removing a point from $E$ will destroy any fibration structure if there was one.


It is a old theorem of Hermann that any Riemannian submersion $E\to B$ with complete $E$
is a locally trivial fiber bundle (whose structure group need not be a Lie group though). Good sources to learn about Riemannian submersions are Besse's book "Einstein manifolds", chapter 9, and Gromoll-Walschap's recent book "Metric folitations and curvature".


