# On quotients of spaces with Ricci curvature bounded below

## TL;DR

This paper proves that synthetic Ricci curvature lower bounds are stable under quotients by isometric group actions and foliations in metric measure spaces, extending classical geometric stability results to a broader synthetic setting.

## Contribution

It establishes the stability of $ ext{RCD}^*(K,N)$-space curvature bounds under quotients, metric foliations, and submetries, generalizing classical Riemannian results to synthetic Ricci curvature bounds.

## Key findings

- Quotients of $ ext{RCD}^*(K,N)$-spaces retain Ricci curvature bounds.
- Extension of classical stability results to synthetic curvature-dimension conditions.
- Applications include classification and rigidity theorems for spaces with Ricci bounds.

## Abstract

Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreover, the analogous stability property holds for metric foliations and submersions. The goal of the paper is to prove the corresponding stability properties for synthetic Ricci curvature lower bounds. Specifically, we show that such stability holds for quotients of $\mathsf{RCD}^{*}(K,N)$-spaces, under isomorphic compact group actions and more generally under metric-measure foliations and submetries. An $\mathsf{RCD}^{*}(K,N)$-space is a metric measure space with an upper dimension bound $N$ and weighted Ricci curvature bounded below by $K$ in a generalized sense. In particular, this shows that if $(M,g)$ has Ricci curvature bounded below by $K\in \mathbb{R}$ and dimension $N$, then the quotient space is an $\mathsf{RCD}^{*}(K,N)$-space. Additionally, we tackle the same problem for the $\mathsf{CD}/\mathsf{CD}^*$ and $\mathsf{MCP}$ curvature-dimension conditions. We provide as well geometric applications which include: A generalization of Kobayashi's Classification Theorem of homogenous manifolds to $\mathsf{RCD}^{*}(K,N)$-spaces with essential minimal dimension $n\leq N$; a structure theorem for $\mathsf{RCD}^{*}(K,N)$-spaces admitting actions by large (compact) groups; and geometric rigidity results for orbifolds such as Cheng's Maximal Diameter and Maximal Volume Rigidity Theorems. Finally, in two appendices we apply the methods of the paper to study quotients by isometric group actions of discrete spaces and of (super-)Ricci flows.