On the Pseudohermitian Curvature of Contact Semi-Riemannian Manifolds

Let M be a contact semi-Riemannian manifold, equivalently a non degenerate almost CR manifold. In this paper we study the pseudo-hermitian Ricci curvature, pseudo-Einstein and $$\eta $$-Einstein manifolds. Then, by using the pseudo-Einstein and the $$\eta $$-Einstein conditions, some rigidity theorems are established to characterize Sasakian manifolds among nondegenerate CR manifolds. In particular, if the Webster metric $$g_\theta $$ of nondegenerate CR structure $$({\mathcal {H}},\theta ,J)$$ is pseudo-Einstein with Webster scalar curvature $${\hat{r}}\ne 0$$, then there exists a real constant $$t\ne 0$$ for which the Webster metric associated to $$({\mathcal {H}},t\theta ,J)$$ is Einstein–Sasakian.