  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.
