On quotients of spaces with Ricci curvature bounded below

2017/04/18 by Fernando Galaz-García, Fernando Galaz‐García, Martin Kell +2 · 33 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Bounded function #Combinatorics #Curvature #Dimension (graph theory) #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Mathematical analysis #Mathematics #Pure mathematics #Quotient #Ricci curvature #Scalar curvature #Sectional curvature #Space (punctuation) #math.DG #math.MG

paper · pdf · doi:10.1016/j.jfa.2018.06.002

published in Journal of Functional Analysis 275(6), 1368-1446 (Elsevier BV) · 51 pages. Comments welcome!

arxiv created 2017/04/18 · openalex publication_date 2018/06/15 · arxiv updated 2019/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Let (M,g) be a smooth Riemannian manifold and 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 RCD^*(K,N)-spaces, under isomorphic compact group actions and more generally under metric-measure foliations and submetries. An 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∈ ℝ and dimension N, then the quotient space is an RCD^*(K,N)-space. Additionally, we tackle the same problem for the CD/CD^* and MCP curvature-dimension conditions. We provide as well geometric applications which include: A generalization of Kobayashi's Classification Theorem of homogenous manifolds to RCD^*(K,N)-spaces with essential minimal dimension n≤ N; a structure theorem for 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.

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• Gradient flows of the entropy for finite Markov chains
• Ricci curvature of Markov chains on metric spaces
• Metriques de Carnot-Caratheodory et Quasiisometries des Espaces Symetriques de rang un
• The Globalization Theorem for the Curvature Dimension Condition

Cited by

• Inradius collapsed manifolds with a lower Ricci curvature bound
• Normal Curves in Sub-Finsler Lie Groups: Branching for Strongly Convex Norms and Face Stability for Polyhedral Norms
• Non-collapsed eGH convergence and dimension
• Classification of foliations via disintegration maps
• Configuration spaces over singular spaces -- I. Dirichlet-Form and Metric Measure Geometry
• Do Riemannian Submersions Preserve Positive Intermediate Ricci Curvature?
• Duality and quotient spaces of generalized Wasserstein spaces

Related

• Invariant measures and lower Ricci curvature bounds
• On the isometry group of RCD^*(K,N)-spaces
• Rigidity of Curvature Bounds of Quotient Spaces Of Isometric Actions
• The isometry group of an RCD^*-space is Lie
• Fibrations, and stability for compact group actions on manifolds with local bounded Ricci covering geometry