It is known that the Dirichlet heat flow in a convex domain of Euclidean space  transmits log-concavity from the initial time to any time. Firstly, I introduce a notion of generalized concavity and characterize a concavity preserved by the Dirichlet heat flow in a convex domain of Euclidean. Secondly, I explain that in a totally convex domain of a Riemannian manifold, if some concavity is preserved by the Dirichlet heat flow, then the sectional curvature must vanish on the domain. The first part is based on joint work with Kazuhiro Ishige and Paolo Salani, and the second part is based on joint work with Kazuhiro Ishige and Haruto Tokunaga.