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Scalar curvature via local extent

Research paper by Giona Veronelli

Indexed on: 19 Oct '17Published on: 19 Oct '17Published in: arXiv - Mathematics - Differential Geometry



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.