  Illustration of the geometric and topological properties of Berry phase is
often in an obscure and abstract language of fiber bundles. In this article, we
demonstrate these properties with a lucid and concrete system whose parameter
space is a torus. The instantaneous eigenstate is regarded as a basis. And the
corresponding connection and curvature are calculated respectively.
Furthermore, we find the magnitude of curvature is exactly the Gaussian
curvature, which shows its local property. The topological property is
reflected by the integral over the torus due to Gauss-Bonnet theorem. When we
study the property of parallel transportation of a vector over a loop, we make
a conclusion that the Berry phase is just the angle between the final and
initial vectors. And we also illuminate the geometric meaning of gauge
transformation, which is just a rotation of basis.
