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?
Best Answer
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 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.