The real Jacobi group $G^J_1(\mathbb{ R})$, defined as the semi-direct product of the group ${ \rm SL}(2,\mathbb{ R})$ with the Heisenberg group $H_1$, is embedded in a $4\times 4$ matrix realisation of the group ${ \rm Sp}(2,\mathbb{ R})$. The left-invariant one-forms on $G^J_1(\mathbb{ R})$ 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 ${ \rm SL}(2,\mathbb{ R})$ is presented. The well known Kähler balanced metric in the variables $(x,y,p,q)$ of the four-dimensional Siegel-Jacobi upper half-plane $\mathcal{ X}^J_1=\frac{ G^J_1(\mathbb{ R})}{ { \rm SO}(2) \times\mathbb{ R}} \approx\mathcal{ X}_1 \times\mathbb{ R}^2$ depending of $(\alpha,\gamma)$ 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.
Categories: math.DG