The coherent state representation of the Jacobi group $G^J_1$ is indexed with two parameters, $\mu (=\frac{ 1}{ \hbar})$, describing the part coming from the Heisenberg group, and $k$, characterizing the positive discrete series representation of $\text{ SU}(1,1)$. The Ricci form, the scalar curvature and the geodesics of the Siegel-Jacobi disk $\mathcal{ D}^J_1$ are investigated. The significance in the language of coherent states of the transform which realizes the fundamental conjecture on the Siegel-Jacobi disk is emphasized. The Berezin kernel, Calabi's diastasis, the Kobayashi embedding, and the Cauchy formula for the Sigel-Jacobi disk are presented.
Categories: math.DG math-ph math.MP