# Evenly submersive maps

A map $f: X\to S$ is called evenly submersive if each $s\in S$ has a neighborhood $W$ such that $p^{-1}W$ is covered by open sets $U\subset X$ diffeomorphic to $V\times W$ with $V$ open in $X_s$, and such that $p$ restricted to $U$ coincides with projection onto the second factor.

What is an example of an evenly submersive map that is not a fiber bundle? In what way do evenly submersive maps differ from fiber bundles?

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It seems that by removing the fiber over a closed disk from a fiber bundle you get an evenly submersive map, right? For a less tautological example, take the double cover of a circle to itself and remove from the source a closed interval that is not just a point. –  damiano Apr 15 '10 at 20:04
What kind of objects are $X$, $S$ and $f$? Given the tags and the word "diffeomorphic", I cannot even make a guess. –  Sergei Ivanov Apr 15 '10 at 20:08