On the role of embeddability in conformal pseudo-hermitian geometry

In this article, we review some recent results about the role of embeddability in conformal CR (Cauchy-Riemann) geometry. We will show how this condition enters in the second variation of the pseudo-hermitian counterpart of the Einstein-Hilbert action, in the positivity of the pseudohermitian mass and in the extremality properties of the CR (Cauchy-Riemann) Yamabe quotient. Given also some explicit examples at hand, this aspect shows sharp differences compared to the Riemannian case.