The movement of a localized pattern appearing after Turing instability is a much attractive topic. In this talk, we consider reaction-diffusion systems which possess localized solutions in two dimensional spaces and investigate the dynamics of the solutions on a two dimensional curved surface. In order to analyze them, we first assume the existence of a linearly stable localized solution in the whole space and consider the movement of the solution when the domain is deformed to a curved surface. By using the center manifold reduction, we can reduce the dynamics to ODE systems expressed by the gradient of curvatures.