In deformation of complex analytic spaces, one usually considers an analytic proper simple surjective map $\varpi: \mathscr{M} \twoheadrightarrow \mathscr{P}$ as an analytic family. However the term simple differs by literature. There are two notions of simple: as a locally trivial map and as a flat map. **Are these definitions equivalent?**

I should mention too that some references define an analytic family using submersions instead of simpleness for the smooth analytic case. But this definition coincides with the first one by Ehresmann's fibration theorem.

Maybe the question is trivial, but since I have never seen it written anywhere, I think I should be more careful.

Thanks in advance.

**EDIT:** The locally triviality is in the category of real differentiable manifolds or the category of topological manifolds. This explains why I have cited Ehresmann fibration theorem as an alternative for the statement.