Motivated by the recent work of Tsai-Tsui-Wang, we study the rigidity of maps between compact manifolds with positive curvature. We show that a distance non-increasing map between complex projective spaces is either an isometry or homotopically trivial. The rigidity result also extends to a broader class of manifolds with positive curvature under a weaker contracting condition on the map, interpolating between distance non-increasing and area non-increasing. This approach is based on the harmonic map heat flow. 