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# A cohomology associated to a Riemannian manifold

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:

\cancel{ \text{Does this sequence of cohomologies detrmine the geometry of $$N$$? Namely is it true to say that two nonisometric metrics on $$N$$ give two different cohomolgy sequence?}}