Abstract. The inverse of the metric matrices on the Siegel-Jacobi upper half space ${\mathcal{X}}^J_n$, invariant to the restricted real Jacobi group $G^J_n(\mathbb{R})_0$ and extended Siegel-Jacobi $\tilde {{\mathcal{X}}}^J_n$ upper half space, invariant to the action of the real Jacobi $G^J_n(\mathbb{R})$, are presented. The results are relevant for Berezin quantization of the manifolds ${\mathcal{X}}^J_ n$ and $\tilde {\mathcal{X}}^J_n$. Explicit calculations in the case $n=2$ are given.

Key words: Berry phase, Berry connection, Connection matrix, Jacobi group, invariant metric, Siegel-Jacobi disk, Siegel-Jacobi upper half plane, extended Siegel-Jacobi upper half plane, almost cosymplectic manifold.