to $(M,J,$ $\theta_{0})$, respectively.

Finally, we recall that

We call a CR structure $J$ spherical if Cartan curvature tensor $Q_{11}$ vanishes identically. Here $$Q_{11}=\frac{1}{6}R_{11}+\frac{i}{2}RA_{11}-A_{11,0}-\frac{2i}{3}A_{11,\overset{\_}{1}1}.$$ Note that $(M,J,\theta)$ is called a spherical CR $3$-manifold if $J$ is a spherical structure. We observe that the spherical structure is CR invariant and a closed spherical CR $3$-manifold $(M,J,\theta)$ is locally CR equivalent to the standard CR $3$-sphere $(\mathbf{S}^{3},\widehat{J},\widehat{\theta}).$ In additional, if $M$ is simply connected, then $(M,J,\theta)$ is the standard CR $3$-sphere.

Proofs of Main Theorems
=======================

In this section, we prove the main theorems. We start from the groundwork for Theorem \[t1\].

\[l2\] If $(M,J,\theta)$ is a strictly pseudoconvex $3$-manifold with $c_{1}(T_{1,0}M)=0$, then there is a pure imaginary $1$-form $$\sigma=\sigma_{\overline{1}}\theta^{\overline{1}}-\sigma_{1}\theta^{1}+i\sigma_{0}\theta$$ with $d\omega_{1}^{1}=d\sigma$ such that$$\left \{
\begin{array}
[c]{l}R=R_{1\overline{1}}=\sigma_{\overline{1},1}+\sigma_{1,\overline{1}}-\sigma
_{0}\\
A_{11,\overline{1}}=\sigma_{1,0}+i\sigma_{0,1}-A_{11}\sigma_{\overline{1}}\end{array}
\right.  .\label{1}$$

Because$$c_{1}(T_{1,0}M)=-\frac{1}{2\pi i}\left[  d\omega_{1}^{1}\right]  =0,$$

we know there is a pure imaginary $1$-form$$\sigma=\sigma_{\overline{1}}\theta^{\overline{1}}-\sigma_{1}\theta^{1}+i\sigma_{0}\theta$$ such that$$d\omega_{1}^{1}=d\sigma.$$

By the structure equation$$\left \{
\begin{array}
[c]{l}d\theta=i\theta^{1}\wedge \theta^{\overline{1}}\\
d\theta^{1}=A_{\overline{1}\overline{1}}\theta \wedge \theta^{1}\end{array}
\right.  ,$$ we have$$\begin{array}
[c]{ccl}d\sigma & = & \left(  \sigma_{\overline{1},1}\theta^{1}+\sigma_{\overline
{1},0}\theta \right)  \wedge \theta^{\overline{1}}+\sigma_{\overline{1}}d\theta^{\overline{1}}-\left(  \sigma_{1,\overline{1}}\theta^{\overline{1}}+\sigma_{1,0}\theta \right)  \wedge \theta^{1}-\\
&  & \sigma_{1}d\theta^{1}+i\left(  \sigma_{0,1}\theta^{1}+\sigma
_{0,\overline{1}}\theta^{\overline{1}}\right)  \wedge \theta+i\sigma_{0}d\theta \\
& = & \left(  \sigma_{\overline{1},1}+\sigma_{1,\overline{1}}-\sigma
_{0}\right)  \theta^{1}\wedge \theta^{\overline{1}}-\left(  \sigma
_{1,0}+i\sigma_{0,1}-\sigma_{\overline{1}}A_{11}\right)  \theta \wedge
\theta^{1}+\\
&  & \left(  \sigma_{\overline{1},0}-i\sigma_{0,\overline{1}}-\sigma
_{1}A_{\overline{1}\overline{1}}\right)  \theta \wedge 