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 $(x,y,θ,p,q,κ)$. We show that the parameter $η$ that appears passing from Perelomov's un-normalized coherent state vector based on the Siegel--Jacobi disk $\mathcal{D}^J_1$ to the normalized one is $η=q+\rm{i} p$. The two-parameter invariant metric on the Siegel--Jacobi upper half-plane $\mathcal{X}^J_1=\frac{G^J_1(\R)}{\rm{SO}(2)\times\mathbb{R}}$ is expressed in the variables $(x,y,\rm{Re}~η,\rm{Im}~η)$. It is proved that the five dimensional manifold $\tilde{\mathcal{X}}^J_1=\frac{G^J_1(\R)}{\rm{SO}(2)}\approx\mathcal{X}^J_1\times\mathbb{R}$, 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 $G^J_1(\mathbb{R})$, and its geodesic vectors are determined.

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