  Let ${\mathbb D}_n $ be the generalized unit disk of degree $n$. In this
paper, we find Riemannian metrics on the Siegel-Jacobi disk ${\mathbb D}_n
\times {\mathbb C}^{(m,n)}$ which are invariant under the natural action of the
Jacobi group explicitly and also compute the Laplacians of these invariant
metrics explicitly. These are expressed in terms of the trace form. We give a
brief remark on the theory of harmonic analysis on the Siegel-Jacobi disk
${\mathbb D}_n \times {\mathbb C}^{(m,n)}$.
