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

Abstract

The real Jacobi group $G_1^J(\mathbb{R})=\mathrm{SL}(2,\mathbb{R})⋉{\mathrm{H}}_1$, where ${\mathrm{H}}_1$ 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 $\eta =q+\mathrm{i}\mathrm{p}$. The two-parameter invariant metric on the Siegel--Jacobi upper half-plane ${\mathcal{X}}_1^J=\frac{G_1^J(\mathbb{R})}{\mathrm{SO}(2)\times \mathbb{R}}$ is expressed in the variables $(x,y,\mathrm{Re}\eta ,\mathrm{Im}\eta )$. It is proved that the five dimensional manifold ${\tilde{\mathcal{X}}}_1^J=\frac{G_1^J(\mathbb{R})}{\mathrm{SO}(2)}\approx {\mathcal{X}}_1^J\times \mathbb{R}$, 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.

26 pages, Latex, amsart, AMS fonts; the abstract and the introduction are improved, some typos are eliminated, more references are added; arXiv admin note: text overlap with arXiv:1903.10721

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• A holomorphic representation of the multidimensional Jacobi algebra
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Cited by in corpus (2)

• Invariant metric on the extended Siegel-Jacobi upper half space
• Hamiltonian systems on almost cosymplectic manifolds