Scalar curvature via local extent
Differential Geometry
2022-12-19 v2 Metric Geometry
Abstract
We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between 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.
Cite
@article{arxiv.1710.07178,
title = {Scalar curvature via local extent},
author = {Giona Veronelli},
journal= {arXiv preprint arXiv:1710.07178},
year = {2022}
}
Comments
22 pages. A new rigidity result has been added (see Proposition 17). Some typos have been corrected