I was wondering whether there is some notion of "vector bundle de Rham cohomology".
To be more precise: the k-th de Rham cohomology group of a manifold $H\_{dR}^{k}(M)$ is defined as the set of closed forms in $\Omega^k(M)$ modulo the set of exact forms. The coboundary operator is given by the exterior derivative.


Let now $E \rightarrow M$ be a vector bundle with connection $\nabla^E$ over $M$, and consider the $E$-valued $k$-forms on $M$: $\Omega^k(M,E)=\Gamma(\Lambda^k TM^\ast \otimes E)$.  

If $E$ is a **flat** vector bundle, we get a coboundary operator $d^{\nabla^E}$ (since $d^{\nabla^E} \circ d^{\nabla^E} = R^{\nabla^E}=0$, with $R^{\nabla^E}$ being the curvature) and we can define 



$$H\_{dR}^{k}(M,E) := \frac{ker \quad d^{\nabla^E}|\_{\Omega^k(M,E)}}{im \quad d^{\nabla^E}|\_{\Omega^{k-1}(M,E)}}$$



So my question: Is this somehow useful? I mean can one use this definition to make some statements about $M$ or $E$ or whatever? Or is the restriction of $E$ to be a flat vector bundle somehow disturbing? Or is this completely useless?



de Rham cohomology and flat vector bundles
Warning: The first paragraph of the following is outside my expertise.


I am told this construction is very useful in PDE's. If you have a PDE on some manifold $M$, you can often formulate the vector space of solutions as the kernel of some flat connection on a vector bundle. In particular, I believe that the analytic side of the [Atiyah-Singer index theorem](http://en.wikipedia.org/wiki/Atiyah-Singer_index_theorem#The_analytical_index) is the Euler characteristic of the deRham theory you have described.


I can tell you that the analogous construction is very important in complex algebraic geometry. Given a holomorphic vector bundle on a complex manifold, there is a natural way to define a $d$-bar connection on it. (This mean $\nabla\_X$ is only defined when $X$ is a $(0,1)$ vector field.) The cohomology of the resulting deRham-like complex, which is called the Dolbeault complex in this setting, is the same as the cohomology of the sheaf of holomorphic sections of the vector bundle. See Wells' *Differential Analysis on Complex Manifolds* or the early parts of Voisin's *Hodge Theory and Algebraic Geometry*.


