The Real Jacobi Group Revisited

The real Jacobi group $G_1^J(\mathbb{R})$, defined as the semi-direct product of the group $\mathrm{S}\mathrm{L}(2,\mathbb{R})$ with the Heisenberg group $H_1$, is embedded in a $4\times 4$ matrix realisation of the group ${\rm Sp}(2,$\mathbb{R}$)$\mathrm{.}Theleft-invariantone-formson$G^J_1($\mathbb{R}$)$ and their dual orthogonal left-invariant vector fields are calculated in the S-coordinates $(x,y,\theta ,p,q,\kappa )$, and a left-invariant metric depending of 4 parameters $(\alpha ,\beta ,\gamma ,\delta )$ is obtained. An invariant metric depending of $(\alpha ,\beta )$ in the variables $(x,y,\theta )$ on the Sasaki manifold $\mathrm{S}\mathrm{L}(2,\mathbb{R})$ is presented. The well known Kähler balanced metric in the variables $(x,y,p,q)$ of the four-dimensional Siegel-Jacobi upper half-plane $$\mathcal{X}$^J_1=\frac{G^J_1($\mathbb{R}$)}{{\rm SO}(2) \times$\mathbb{R}$} \approx$\mathcal{X}$_1 \times$\mathbb{R}$^2$dependingof$(\alpha,\gamma)$ is written down as sum of the squares of four invariant one-forms, where ${\mathcal{X}}_1$ denotes the Siegel upper half-plane. The left-invariant metric in the variables $(x,y,p,q,\kappa )$ depending on $(\alpha ,\gamma ,\delta )$ of a five-dimensional manifold $\tilde{$\mathcal{X}$}^J_1= \frac{G^J_1($\mathbb{R}$)}{{\rm SO}(2)}\approx$\mathcal{X}$_1\times$\mathbb{R}$^3$ is determined.