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

A. G. Sedykh

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 5 (2015), pp. 48-55

Résumé

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, $\eta=\theta^3$, $\varphi_0=\begin{pmatrix}0-10\\ 100\\ 000\end{pmatrix}$, 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 $$ \eta=\theta^3, \quad \xi=e_3, \quad, \varphi= \begin{pmatrix} \frac{2\rho\sin\alpha_1}{-1+\rho^2} \frac{-1+2\rho\cos\alpha_1-\rho^2}{1-\rho^2} 0\\ \frac{1+2\rho\cos\alpha_1+\rho^2}{1-\rho^2} \frac{2\rho\sin\alpha_1}{1-\rho^2} 0\\ 0 0 1 \end{pmatrix}. $$

Détail

Mots-clés : Lie group
Keywords: contact form, contact metric structure.