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

Abstract

We show that on every $\mathsf{RCD}$ 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 $\mathsf{RCD}$ spaces, our construction applies in particular to the Alexandrov setting. We conjecture that an $\mathsf{RCD}$ space is Alexandrov if and only if the sectional curvature - defined in terms of such abstract Riemann tensor - is bounded from below.

References in corpus (2)

• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
• Semiconcave functions in Alexandrov's geometry