On Scalar and Ricci Curvatures

Gérard Besson (1) , S Gallot (1)

1 IF - Institut Fourier

The purpose of this report is to acknowledge the influence of M. Gromov's vision of Geometry on our own work. 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 measured spaces. We describe recent results some of which are already posted in [BCGS20b] where we proposed to use the volume entropy. We also attempt to give a new synthetic version of Ricci curvature boundedbelow using Bishop-Gromov's Inequality.

Mots clés: scalar curvature; Whitehead manifold; metric measured space; volume entropy; Ricci curvature; Bishop-Gromov's Inequality

Domaines: Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]; Théorie des groupes [math.GR]

Gérard Besson, S Gallot. On Scalar and Ricci Curvatures. [Research Report] Université Grenoble Alpes [2020-..]; CNRS-UMR 5582 Institut Fourier. 2020. ⟨hal-02966630v1⟩