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

Nicola Gigli

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arXiv · 1902.02282

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

Nicola Gigli

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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.

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Nicola Gigli. 2019-02-06. Riemann curvature tensor on ${\sf RCD}$ spaces and possible applications. https://arxiv.org/abs/1902.02282

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