theorem '
---

Introduction
============

 In Riemannian geometry, a Riemannian manifold is called Einstein if the Ricci curvature tensor is function-proportional to its Riemannian metric. For dimension greater than $2$, it is equivalent to the constant-proportional case. In the contrast to the Riemannian geometry situation, there is a resembling notion that a strictly pseudoconvex CR $(2n+1)$-manifold is called pseudo-Einstein if the pseudohermitian Ricci curvature tensor is function-proportional to its Levi metric. The pseudo-Einstein condition is less rigid than the Einstein condition in Riemannian geometry. Indeed, the CR contracted Bianchi identity no longer implies the pseudohermitian scalar curvature $R$ to be a constant due to the presence of pseudohermitian torsion for $n\geq2$ $$R_{\alpha \overline{\beta}}{}_{,\beta}=R_{\alpha}-i(n-1)A_{\alpha \beta}{}_{,\overline{\beta}}.$$ Note that any contact form on a closed strictly pseudoconvex $3$-manifold is actually pseudo-Einstein since the pseudohermitian Ricci tensor has only one component $R_{1\overline{1}}$.

In [@l], J. Lee showed that the obstruction to the existence of a pseudo-Einstein contact form $\theta$ is that its first Chern class $c_{1}(T_{1,0}M)$ vanishes. Indeed, for a closed strictly pseudoconvex $(2n+1)
$-manifold $\left(  M,J,\theta \right)  $ with $c_{1}(T_{1,0}M)=0$ and $n\geq2
$, he proved that $M$ admits a globally defined pseudo-Einstein contact form if either $M$ admits a contact form $\theta$ with 