Suppose $p: X \rightarrow S$ is a fiber bundle of smooth manifold, if the Gauss-Manin connection is nontrivial, could $p$ be trivial bundle as smooth manifold? Also, could $p$ be trivial bundle as topological manifold?
The case I am interested most is when $X$ and $S$ are complex manifolds. Feel free to restrict to this case if it is easier.