\vert )$$ and the CR $Q$-curvature is vanishing $$\Delta_{b}R-i(A_{11,\bar{1}\bar{1}}-A_{\bar{1}\bar{1},11})=0.$$ Then $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form for any CR-pluriharmonic function $f.$

\[c6\] Let $(M,J,\theta)$ is a closed strictly pseudoconvex CR $3$-manifold of $c_{1}(T_{1,0}M)=0$ with nonnegative CR Paneitz operator $P_{0}. $ Assume that  the pseudohermitian curvature is $\left(  \frac{1}{2},\frac{1}{2}\right)  $-positive and the CR $Q$-curvature is vanishing. Then $(M,J,\theta)$ is a pseudo-Einstein CR manifold. In additional if it is spherical with positive constant Tanaka-Webster scalar curvature, then $(M,J,\theta)$ is the Sasakian space form with the positive constant Tanaka-Webster scalar curvature and vanishing torsion.

We conclude this introduction with a brief plan of the paper. In Section $2$, we derive some preliminary results and indicate the geometry and topology of CR $3$-manifolds with the positivity of pseudohermitian curvature. In Section $3,\ $we prove main Theorems. In Appendix, we survey basic notions in the pseudohermitian (strictly pseudoconvex CR) geometry.

**Acknowledgements** Part of the project was done during the visit to Yau Mathematical Sciences Center, Tsinghua University. The authors would like to express their thanks for the warm hospitality.

Preliminaries
=============

In this section, we derive some necessary ingredients for the proof of main results in this paper. In particular, we define the 