3 fixed spelling in the title

# de RahmRham cohomology and flat vector bundles

2 Added p tag around displayed maths

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 $d^{\nabla^E} = R^{\nabla^E}=0$, with $R^{\nabla^E}$ being the curvature) and we can define

$$H_{dR}^{k}(M,E) := \frac{ker \quad d^{\nabla^E}|{\Omega^k(M,E)}}{im d^{\nabla^E}|_{\Omega^k(M,E)}}{im \quad d^{\nabla^E}|{\Omega^{k-1}(M,E)}}$$d^{\nabla^E}|_{\Omega^{k-1}(M,E)}}$$So my question: Is this somehow useful? I mean can one use this definition to make some statements about M or E or whatever? Or is the restriction of E to be a flat vector bundle somehow disturbing? Or is this completely useless? 1 # de Rahm cohomology and flat vector bundles I was wondering whether there is some notion of "vector bundle de Rham cohomology". To be more precise: the k-th de Rham cohomology group of a manifold H_{dR}^{k}(M) is defined as the set of closed forms in \Omega^k(M) modulo the set of exact forms. The coboundary operator is given by the exterior derivative. Let now E \rightarrow M be a vector bundle with connection \nabla^E over M, and consider the E-valued k-forms on M: \Omega^k(M,E)=\Gamma(\Lambda^k TM^\ast \otimes E). If E is a flat vector bundle, we get a coboundary operator d^{\nabla^E} (since d^{\nabla^E} = R^{\nabla^E}=0, with R^{\nabla^E} being the curvature) and we can define$$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?