Scalar curvature via local extent

TL;DR

This paper provides a metric-based characterization of scalar curvature using local point configurations, enabling potential extension to non-smooth spaces like Alexandrov spaces and surfaces with bounded curvature.

Contribution

It introduces a new metric approach to defining scalar curvature through local extent, applicable to non-smooth metric spaces.

Findings

Characterization of scalar curvature via local extent in Riemannian manifolds

Potential extension of scalar curvature definition to non-smooth metric spaces

Discussion on scalar curvature in Alexandrov spaces and surfaces with bounded integral curvature

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.