Abstract: In this talk, we are going to consider the rigidity of map between positively curved closed manifolds, which is motivated by the recent work of Tsai-Tsui-Wang. We show that distance non-increasing map between complex projective spaces is either an isometry or homotopically trivial. The rigidity result also holds on a wider class of manifolds with positive curvature and weaker contracting property on the map in between distance non-increasing and area non-increasing. This is based on the harmonic map heat flow and it partially answer a question raised by Tsai-Tsui-Wang. This is a joint work with Man-Chun Lee.