MATHEMATICS

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

Ya. V. Slavolyubova

Kemerovo Institute (branch) of Plekhanov Russian University of Economics, Kemerovo, Russian Federation

Abstract: 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 $\varphi_0=\begin{pmatrix}J_0&0\\0&0\end{pmatrix}$ and the metric $g_0=\begin{pmatrix}I&0\\0&\lambda\end{pmatrix}$ 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 $P\mid_{\mathrm{ker}\,\eta}=\begin{pmatrix}A&B&D\\ B&C&F\\ D&F&N\end{pmatrix}$, where $A=\begin{pmatrix}u&v\\ v&-u\end{pmatrix}$, $B=\begin{pmatrix}s&t\\ t&-s\end{pmatrix}$, $C=\begin{pmatrix}k& l\\ l& -k\end{pmatrix}$, $D=\begin{pmatrix}x & y\\ y& -x\end{pmatrix}$, $F=\begin{pmatrix}q& r\\ r& -q\end{pmatrix}$, 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
$$ P\mid_{\mathrm{ker}\,\eta}=\begin{pmatrix}0&B&0\\ B&0&0\\ 0&0&0\end{pmatrix}, P\mid_{\mathrm{ker}\,\eta}=\begin{pmatrix}0&0& D\\ 0&0&0\\ D&0&0\end{pmatrix}, P\mid_{\mathrm{ker}\,\eta}=\begin{pmatrix}0&0&0\\ 0&0& F\\ 0&F&0\end{pmatrix}, P\mid_{\mathrm{ker}\,\eta}=\begin{pmatrix}A&0&0\\ 0&C&0\\ 0&0& N\end{pmatrix}. $$

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:
$$ A_{Ric}=\begin{pmatrix}-\frac\lambda2& 0&0&0&0&0&0\\ 0&-\frac\lambda2&0&0&0&0&0\\ 0&0&-\frac\lambda2&0&0&0&0\\ 0&0&0&-\frac\lambda2&0&0&0\\ 0&0&0&0&-\frac\lambda2&0&0\\ 0&0&0&0&0&-\frac\lambda2&0\\ 0&0&0&0&0&0&-\frac\lambda2\end{pmatrix}. $$

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 $Ric_{g_0}(X,Y)=-\frac\lambda2g_0(X,Y)+\frac{(n+\lambda)\lambda}2\eta(X)\eta(Y)$, $X,Y\in L(H^{2n+1})$.

Keywords: Lie group, contact metric structures, associated metric.

UDC: 514.76

MSC: 53D10

Received: 27.02.2018

DOI: 10.17223/19988621/54/3

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