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Let $E$ be a smooth vector bundle on a smooth manifold $M$. If $E$ is flat, there is a connection $\nabla$ which is a differential which we can use to define the de Rham cohomology of $E$.

If $E$ is a non-flat bundle on $M$, is there any meaningful way to construct a de Rham cohomology of $E$?

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    @GunnarMagnusson: I was thinking of a cohomology theory defined using forms. I know that for a non-flat bundle you can't use a connection to define a complex, but I was wondering whether there is a way to twist a connection that would allow for such an approach. Alternatively, I imagine that you could use derived categories to define a cohomology theory for non-flat bundles, I'm just not sure how it would manifest itself geometrically.2012-10-02

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