  The relation between the Berry phase and connection matrix on the
Siegel-Jacobi disk $\mathcal{D}^J_1$ and Siegel-Jacobi upper
half-plane$\mathcal{X}^J_1$ are analyzed. The connection matrix and the
covariant derivative of one-forms on the extended Siegel-Jacobi upper
half-plane $\tilde{\mathcal{X}}^J_1$ are calculated.
