Calculus of Variations and Geometric Measure Theory
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N. Gigli

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

created by gigli on 20 Jul 2020



Inserted: 20 jul 2020

Year: 2019

ArXiv: 1902.02282 PDF


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.

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