Abstract: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
$$
\eta=\theta^3, \quad \xi=e_3, \quad,
\varphi=
\begin{pmatrix}
\frac{2\rho\sin\alpha_1}{-1+\rho^2} & \frac{-1+2\rho\cos\alpha_1-\rho^2}{1-\rho^2} & 0\\
\frac{1+2\rho\cos\alpha_1+\rho^2}{1-\rho^2} & \frac{2\rho\sin\alpha_1}{1-\rho^2} & 0\\
0& 0& 1
\end{pmatrix}.
$$
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