On Scalar and Ricci Curvatures

The purpose of this report is to acknowledge the influence of M. Gromov's vision of geometry on our own works. It is two-fold: in the first part we aim at describing some results, in dimension 3, around the question: which open 3-manifolds carry a complete Riemannian metric of positive or non negative scalar curvature? In the second part we look for weak forms of the notion of “lower bounds of the Ricci curvature” on non necessarily smooth metric measure spaces. We describe recent results some of which are already posted in [arXiv:1712.08386] where we proposed to use the volume entropy. We also attempt to give a new synthetic version of Ricci curvature bounded below using Bishop–Gromov's inequality.