Level set approach has been considered as a powerful topology optimization (TO) tool for generating innovative designs,but the conventional level set framework is built in the Euclidean space (R2 and R3),which can not satisfy the increasing demands of topology optimization on free-form surfaces.In this paper,we propose a new method to address the problem of topology optimization for conformal structures on freeform surfaces which are mathematically termed as manifolds.By using conformal parametrization,we extend the conventional level-set-based topology optimization framework from Euclidean space (R2 and R3) to surface with arbitrary topologies.