We discuss the preservation of concavity under the porous medium flow on a Riemannian manifold,  and clarify the roles of the diffusion nonlinearity and the curvature.
In particular,  we establish curvature obstructions: If the sectional curvature is nonpositive and its minimum is negative at some point, then any concavity is not preserved. Furthermore, if the sectional curvature is nonzero somewhere, then any preserved concavity must be stronger than the concavity associated with the diffusion nonlinearity. This talk is based on joint work with Kazuhiro Ishige and Yoshiumi Tateoka.