Berry phases and connection matrices defined on homogeneous spaces attached to Siegel-Jacobi groups

The relation between the Berry phase and connection matrix on the Siegel-Jacobi disk ${\mathcal{D}}_1^J$ and Siegel-Jacobi upper half-plane${\mathcal{X}}_1^J$ are analyzed. The connection matrix and the covariant derivative of one-forms on the extended Siegel-Jacobi upper half-plane ${\tilde{\mathcal{X}}}_1^J$ are calculated.