  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
K\"ahler manifolds.
