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

## TL;DR

This paper develops a way to define a Riemann curvature tensor on ${\sf RCD}$ spaces using a distributional approach, linking curvature bounds to Alexandrov spaces and proposing a conjecture about their equivalence.

## Contribution

It introduces a novel distributional method to define the Riemann curvature tensor on ${\sf RCD}$ spaces, extending geometric analysis tools to this setting.

## Key findings

- Construction of a distributional Riemann curvature tensor on ${\sf RCD}$ spaces
- Application of the construction to Alexandrov spaces due to their ${\sf RCD}$ property
- Conjecture relating sectional curvature bounds to Alexandrov space characterization

## 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.