Associated left-invariant contact metric structures on the $7$-dimensional Heisenberg group~$H^7$

Ya. V. Slavolyubova

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 54 (2018), pp. 34-45

Résumé

In this paper, we construct new nonstandard associated left-invariant contact metric structures $(\eta,\xi,\varphi,g_\lambda)$ on the $7$-dimensional Heisenberg group $H^7$. The associated left-invariant contact metric structures for the contact structure $\eta$ on the contact Lie group $(H^7,\eta)$ were given by the affinor $\varphi$ and the (pseudo-)Riemannian metric $g_\lambda$ such that \begin{gather} \varphi\mid_{\mathrm{ker}\,\eta}=J,\quad \varphi(\xi)=0, \notag\\ g_\lambda(X,Y)=d\eta(\varphi X,Y)+\lambda\eta(X)\eta(Y), \end{gather} where $J$ is an almost complex structure compatible with the restriction of $g_\lambda$ on $\mathrm{ker}\,\eta$, $g_\lambda\mid_{\mathrm{ker}\,\eta}$. The parameter $\lambda$ provided deformation of the associated metric $g_\lambda$ along the Reeb field $\xi$. The affinor $\varphi_0=\begin{pmatrix}J_00\\00\end{pmatrix}$ and the metric $g_0=\begin{pmatrix}I0\\0\lambda\end{pmatrix}$ are fixed. The new affinors $\varphi=\varphi_0(Id+P)(Id-P)^{-1}$ are given by an operator $P:L(H^7)\to L(H^7)$ such that $P(\xi)=0$ and $P\mid_{\mathrm{ker}\,\eta}=\begin{pmatrix}A\\ B\\ D\end{pmatrix}$, where $A=\begin{pmatrix}u\\ v-u\end{pmatrix}$, $B=\begin{pmatrix}s\\ t-s\end{pmatrix}$, $C=\begin{pmatrix}k l\\ l -k\end{pmatrix}$, $D=\begin{pmatrix}x y\\ y -x\end{pmatrix}$, $F=\begin{pmatrix}q r\\ r -q\end{pmatrix}$, and $N=\begin{pmatrix}w z\\ z -w\end{pmatrix}$ are symmetric matrices; $u, v, s, t, k, l, x, y, q, r, w$, and $z$ are real parameters. Each new affinor $\varphi$ defines a new associated metric $g_\lambda$ by formula (1). We have considered some particular classes of associated metrics corresponding to the affinors $\varphi$ which were given by the operators $P$ of the following types $$ P\mid_{\mathrm{ker}\,\eta}=\begin{pmatrix}00\\ B00\\ 000\end{pmatrix}, P\mid_{\mathrm{ker}\,\eta}=\begin{pmatrix}00 D\\ 000\\ D00\end{pmatrix}, P\mid_{\mathrm{ker}\,\eta}=\begin{pmatrix}000\\ 00 F\\ 00\end{pmatrix}, P\mid_{\mathrm{ker}\,\eta}=\begin{pmatrix}A00\\ 00\\ 00 N\end{pmatrix}. $$ The following theorem was received for any associated (pseudo-)Riemannian metric $g_\lambda(X,Y)=d\eta(\varphi X,Y)+\lambda\eta(X)\eta(Y)$. Theorem 1. Any left-invariant contact metric structure $(\eta,\xi,\varphi,g_\lambda)$ on the Heisenberg group $H^7$ is a Sasaki, $K$-contact, and $\eta$-Einstein structure. The squares of the norms of a Riemann tensor $R$ and Ricci tensor $Ric(X,Y)=g_\lambda(A_{Ric}X,Y)$ of associated left-invariant metric $g_\lambda$ have the following expressions: $||R||^2=\frac{69\lambda^2}4$, $||Ric||^2=\frac{15\lambda^2}4$. The Ricci operator has the following matrix: $$ A_{Ric}=\begin{pmatrix}-\frac\lambda2 000000\\ 0-\frac\lambda200000\\ 00-\frac\lambda20000\\ 000-\frac\lambda2000\\ 0000-\frac\lambda200\\ 00000-\frac\lambda20\\ 000000-\frac\lambda2\end{pmatrix}. $$ The sign of the scalar curvature of associated left-invariant metric $g_\lambda$ is not constant and $S=-\frac{3\lambda}2$. In addition, the following theorem has been proved for any $(2n+1)$-dimensional Heisenberg group $H^{2n+1}$ with a given (pseudo-)Riemannian metric $g_0=e_1^{^*2}+\dots+e_{2n}^{^*2}+\lambda e_{2n+1}^{^*2}$. Theorem 2. A left-invariant contact metric structure $(\eta,\xi,\varphi_0,g_0)$ on the Heisenberg group $H^{2n+1}$ is $\eta$-Einstein, and $Ric_{g_0}(X,Y)=-\frac\lambda2g_0(X,Y)+\frac{(n+\lambda)\lambda}2\eta(X)\eta(Y)$, $X,Y\in L(H^{2n+1})$.

Détail

Mots-clés : Lie group
Keywords: contact metric structures, associated metric.