pseudoconvex CR $3$-manifold $(M,\theta)$ is said to be pseudo-Einstein if the following tensor is vanishing $$W_{1}\doteqdot \left(  R_{1}-iA_{11_{,}\overline{1}}\right)  =0.$$ (iii) We define the first Chern class $c_{1}(T_{1,0}M)\in H^{2}(M,\mathbf{R})
$ for the holomorphic tangent bundle $T^{1,0}M$ by $$\begin{aligned}
c_{1}(T^{1,0}M)  & =\frac{i}{2\pi}[d\omega_{\alpha}{}^{\alpha}]\\
& =\frac{i}{2\pi}[R_{\alpha \overline{\beta}}\theta^{\alpha}\wedge
\theta^{\overline{\beta}}+A_{\alpha \mu,\overline{\alpha}}{}\theta^{\mu}\wedge \theta-A_{\overline{\alpha}\overline{\mu},\alpha}\theta^{\overline{\mu}}\wedge \theta].\end{aligned}$$ (iv) Note that any pseudo-Einstein manifold $(M^{2n+1},\theta)$, the first Chern class $c_{1}(T_{1,0}M)$ of $T_{1,0}(M)$ is vanishing ([@l]).

Next let us recall the equivalent definitions of the pseudo-Einsteinian $(2n+1)$-manifold for $n\geq2$ and $n=1$ as well.

\[l1\] (i) If $(M,J,\theta)$ is a strictly pseudoconvex CR $(2n+1)$-manifold for $n\geq2$, then the following propositions are all equivalent :$$\begin{array}
[c]{cl}\left(  1\right)  & R_{\alpha \overline{\beta}}=\frac{R}{n}h_{\alpha
\overline{\beta}},\text{ \  \ }\\
\left(  2\right)  & \left(  \omega_{\alpha}^{\alpha}+\frac{i}{n}R\theta \right)  \text{ }\mathrm{is}\text{ }\mathrm{closed,}\\
\left(  3\right)  & W_{\alpha}\doteqdot \left(  R,_{\alpha}-inA_{\alpha \beta
},^{\beta}\right)  =0\text{.\ }\end{array}$$ As for $n=1,$ we still have the equivalent between $(2)$ and $(3).$

\(ii) By the equivalence of $\left(  2\right)  $ and $\left(  3\right)  $, we see the first Chern class $c_{1}(T_{1,0}M)$ is vanishing if $(M,J,\theta)$ is a pseudo-Einsteinian $3$-manifold.

The equivalence of $\left(  1\right)  $ and $\left(  2\right)  $ could be found in [@l] for $n\geq2$. The proof of $\left(  2\right)
\Longleftrightarrow \left(  3\right)  $ for $n\geq2$ is the same with 