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A note on vanishing of equivariant differentiable cohomology of proper actions and application to CR-automorphism and conformal groups

We establish that for any proper action of a Lie group on a manifold the associated equivariant differentiable cohomology groups with coefficients in modules of $\mathcal{C}^\infty$-functions vanish in all degrees except than zero. Furthermore let $G$ be a Lie group of $CR$-automorphisms of a strictly pseudo-convex $CR$-manifold $M$. We associate to $G$ a canonical class in the first differential cohomology of $G$ with coefficients in the $\mathcal{C}^\infty$-functions on $M$. This class is non-zero if and only if $G$ is essential in the sense that there does not exist a $CR$-compatible strictly pseudo-convex pseudo-Hermitian structure on $M$ which is preserved by $G$. We prove that a closed Lie subgroup $G$ of $CR$-automorphisms acts properly on $M$ if and only if its canonical class vanishes. As a consequence of Schoen's theorem, it follows that for any strictly pseudo-convex $CR$-manifold $M$, there exists a compatible strictly pseudo-convex pseudo-Hermitian structure such that the CR-automorphism group for $M$ and the group of pseudo-Hermitian transformations coincide, except for two kinds of spherical $CR$-manifolds. Similar results hold for conformal Riemannian and Kähler manifolds.