Geodesics on the extended Siegel-Jacobi upper half-plane

Abstract

The semidirect product of the real Heisenberg group ${\mathrm{H}}_1(\mathbb{R})$ with $\mathrm{SL}(2,\mathbb{R})$, called the real Jacobi group $G_1^J(\mathbb{R})$, admits a four-parameter invariant metric expressed in the S-coordinates. We determine the geodesic equations on the extended Siegel--Jacobi upper half-plane ${\tilde{\mathcal{X}}}_1^J=\frac{G_1^J(\mathbb{R})}{\mathrm{SO}(2)}\approx {\mathcal{X}}_1^J\times \mathbb{R}\approx {\mathcal{X}}_1\times {\mathbb{R}}^3$, where ${\mathcal{X}}_1^J$ (${\mathcal{X}}_1)$ 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 ${\tilde{\mathcal{X}}}_1^J$, we get the geodesic equations on ${\mathcal{X}}_1^J$, ${\mathcal{X}}_1$ and ${\mathrm{H}}_1(\mathbb{R})$.

27 pages, Latex, amsart, AMS fonts

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Cited by in corpus (1)

• Hamiltonian systems on almost cosymplectic manifolds