Contact metric structures on 3-dimentional non-unimodular Lie groups

Definition 1. A differentiable $(2n+1)$-dimensional manifold $M$ of the class $C^\infty$ is called a contact manifold if there exists a differential $1$-form $\eta$ on $M^{2n+1}$, such that $(\eta\land d\eta)^n\ne0$. The form $\eta$ is called a contact form.
Definition 2. If $M^{2n+1}$ is a contact manifold with a contact form $\eta$, then a contact metric structure is the quadruple $(\eta,\xi,\varphi,g)$, where $\xi$ is a Reeb’s field, $g$ is a Riemannian metric, and $\varphi$ is an affinor on $M^{2n+1}$, for which the following properties are valid:

• $\varphi^2=-I+\eta\otimes\xi$,
• $d\eta(X,Y)=g(X,\varphi Y)$,
• $g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y)$.

We consider a non-unimodular Lie group $G$; its Lie algebra has a basis $e_1$, $e_2$, $e_3$ such that $[e_1,e_2]=\alpha e_2+\beta e_3$, $[e_1,e_3]=\gamma e_2+\delta e_3$, $[e_2,e_3]=0$, and matrix $A=\begin{pmatrix}\alpha & \beta\\ \gamma & \delta\end{pmatrix}$ has a trace $\alpha+\delta=2$.
The left invariant $1$-form $\eta=a_1\theta^1+a_2\theta^2+a_3\theta^3$ defines a contact structure on the group $G$ if $(\delta-\alpha)a_2a_3-\beta a_3^2+\gamma a_2^2\ne0$.
As a contact form, we choose the simplest one, $\eta=\theta^3$, $\varphi_0=\begin{pmatrix}0&-1&0\\ 1&0&0\\ 0&0&0\end{pmatrix}$, and consider other metrics that also define a contact metric form.
We obtain that a contact metric structure on a non-unimodular Lie group can be set by the quadruple $(\eta,\xi,\varphi,g)$, where
$$ \eta=\theta^3, \quad \xi=e_3, \quad, \varphi= \begin{pmatrix} \frac{2\rho\sin\alpha_1}{-1+\rho^2} & \frac{-1+2\rho\cos\alpha_1-\rho^2}{1-\rho^2} & 0\\ \frac{1+2\rho\cos\alpha_1+\rho^2}{1-\rho^2} & \frac{2\rho\sin\alpha_1}{1-\rho^2} & 0\\ 0& 0& 1 \end{pmatrix}. $$