  In this note we show the existence of the hyperbolical geometric quantum
phase that is different from the ordinary trigonometric geometric quantum
phase. Gravitomagnetic charge (dual mass) is the gravitational analogue of
magnetic monopole in Electrodynamics; but, as will be shown here, it possesses
more interesting and significant features, {\it e.g.}, it may constitute the
dual matter that has different gravitational properties compared with mass. In
order to describe the space-time curvature due to the topological dual mass, we
construct the dual Einstein's tensor. Further investigation shows that
gravitomagnetic potentials caused by dual mass are respectively analogous to
the trigonometric and hyperbolic geometric phase. The study of the geometric
phase and dual mass provides a valuable insight into the time evolution of
quantum systems and the topological properties in General Relativity.
