Geodesics on the extended Siegel-Jacobi upper half-plane

The semidirect product of the real Heisenberg group ${\rm H}_1(\mathbb{R})$ with ${\rm SL}(2,\mathbb{R})$, called the real Jacobi group $G^J_1(\mathbb{R})$, admits a four-parameter invariant metric expressed in the S-coordinates. We determine the geodesic equations on the extended Siegel--Jacobi upper half-plane $\tilde{\mathcal{X}}^J_1 =\frac{G^J_1(\R)}{\rm{SO}(2)}\approx\mathcal{X}^J_1\times\mathbb{R}\approx \mathcal{X}_1 \times\mathbb{R}^3$, where $\mathcal{X}^J_1$ ($\mathcal{X}_1)$ denotes the Siegel-Jacobi upper half-plane (respectively Siegel upper half-plane). Equating successively with zero the values of the three parameters in the geodesic equations on $\tilde{\mathcal{X}}^J_1$, we get the geodesic equations on $\mathcal{X}^J_1$, $\mathcal{X}_1$ and ${\rm H}_1(\mathbb{R})$.