Calculus of Variations and Geometric Measure Theory

N. Gigli

Riemann curvature tensor on ${\sf RCD}$ spaces and possible applications

created by gigli on 20 Jul 2020

[BibTeX]

preprint

Inserted: 20 jul 2020

Year: 2019

ArXiv: 1902.02282 PDF

Abstract:

We show that on every ${\sf RCD}$ spaces it is possible to introduce, by a distributional-like approach, a Riemann curvature tensor. Since after the works of Petrunin and Zhang-Zhu we know that finite dimensional Alexandrov spaces are ${\sf RCD}$ spaces, our construction applies in particular to the Alexandrov setting. We conjecture that an ${\sf RCD}$ space is Alexandrov if and only if the sectional curvature - defined in terms of such abstract Riemann tensor - is bounded from below.