Scalar curvature via local extent

Abstract

We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between $(n+1)$ points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of defining the scalar curvature on a non-smooth metric space. In the second part we will discuss this issue, focusing in particular on Alexandrov spaces and surfaces with bounded integral curvature.

22 pages. A new rigidity result has been added (see Proposition 17). Some typos have been corrected

References in corpus (2)

• A stability version of Hölder's inequality
• Metric-measure boundary and geodesic flow on Alexandrov spaces