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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.

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    $\begingroup$ No. If $p$ is a trivial bundle (topologically), the flat bundle $R^ip_*(\mathbb{C})$ is trivial, and so is the Gauss-Manin connection. $\endgroup$
    – abx
    Apr 14, 2019 at 11:06

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