Remarks on the geometry of the extended Siegel--Jacobi upper half-plane

The real Jacobi group $G^J_1($\mathbb{R}$)={\rm SL}(2,$\mathbb{R}$)\ltimes {\rm H}_1$,where${\rm H}_1$ denotes the 3-dimensional Heisenberg group, is parametrized by the $S$-coordinates $(x,y,\theta ,p,q,\kappa )$. We show that the parameter $\eta$ that appears passing from Perelomov’s un-normalized coherent state vector based on the Siegel–Jacobi disk ${\mathcal{D}}_1^J$ to the normalized one is $\eta =q+\mathrm{i}p$. The two-parameter invariant metric on the Siegel–Jacobi upper half-plane ${\mathcal{X}}_1^J=\frac{G_1^J(\mathbb{R})}{\mathrm{S}\mathrm{O}(2)\times \mathbb{R}}$ is expressed in the variables $(x,y,\mathrm{R}\mathrm{e}\eta ,\mathrm{I}\mathrm{m}\eta )$. It is proved that the five dimensional manifold ${\tilde{\mathcal{X}}}_1^J=\frac{G_1^J(\mathbb{R})}{\mathrm{S}\mathrm{O}(2)}\approx {\mathcal{X}}_1^J\times \mathbb{R}$, called extended Siegel–Jacobi upper half-plane, is a reductive, non-symmetric, non-naturally reductive manifold with respect to the three-parameter metric invariant to the action of $G_1^J(\mathbb{R})$, and its geodesic vectors are determined.