Por Jean-Pierre Bourguignon (Nicolaas Kuiper Honorary Professor at the Institut des Hautes Études Scientifiques, Université Paris-Saclay).
The scalar curvature is the weakest invariant involving the curvature of a Riemannian metric. On surfaces, where the concept of curvature was first developed by Carl-Friedrich GAUSS, the curvature reduces to it, but in higher dimensions this scalar function misses a lot of information about the curvature which is a 4-tensor field (it has 20 components in dimension 4).
Still, in the last 60 years problems connected to it have generated a huge amount of literature because of an a priori totally unexpected deep interplay of the existence of a metric with positive scalar curvature with the topology of manifolds.
This has mobilised many radically new approaches, involving in particular spinors and a deeper understanding of a number of topological or differentiable invariants or constructions.
There are still a number of open problems connected to prescribing the scalar curvature on a manifold, and some will be presented.
