The solutions of covariant derivative equations of cross section in associated bundles

A Hopf bundle, whose base manifold is the ring surface T²$T^2$ and whose fiber is the group U(1)$U\left(1\right)$, is established in this paper. On this Hopf bundle, the lifting of the Laplace operator on the base manifold is proved to be the Laplace operator on the Hopf bundle. The solutions of covariant derivative equations of cross section in associated bundles and the index theorem on the ring surface are also discussed.