Let $N$ be a compact Riemanian manifold and $G$ be its isometry group. Let $M=\chi^{\infty}(N)$ be the space of smooth vector fields on $N$. There is a natural right action of $G$ on $M$ with $X.g=g^*(X),\; g\in G, X\in M$, the push forward of $X$ under the isometry $g$.So $M$ is a $G$ module.

How can one express the group cohomologies $H^n(G,M)$ explicitely?Is there a reference which contain such computation?

Edit: according to the comment of Neal I understand the following part of the previous version is a trivial question:

Does this sequence of cohomologies determine the geometry of $N$? Namely is it true to say that two nonisometric metrics on $N$ give two different cohomology sequence?