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Illustration of the geometric and topological properties of Berry phase isoften in an obscure and abstract language of fiber bundles. In this article, wedemonstrate these properties with a lucid and concrete system whose parameterspace is a torus. The instantaneous eigenstate is regarded as a basis. And thecorresponding connection and curvature are calculated respectively.Furthermore, we find the magnitude of curvature is exactly the Gaussiancurvature, which shows its local property. The topological property isreflected by the integral over the torus due to Gauss-Bonnet theorem. When westudy the property of parallel transportation of a vector over a loop, we makea conclusion that the Berry phase is just the angle between the final andinitial vectors. And we also illuminate the geometric meaning of gaugetransformation, which is just a rotation of basis.

The geometric and topological interpretation of Berry phase on a torus