On quotients of spaces with Ricci curvature bounded below

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.

Invariant Measures and Lower Ricci Curvature Bounds

“…Let us denote by n * the dimension of the Euclidean space contained in Tan(X/G p , G p ·q). By Theorem 6.3 of [17] and the previous Lemma k = dim G p · q + n * ≥ k − 1 + n * . Hence n * ≤ 1.…”

“…In [17] some extra assumptions were made in order to obtain information about the behaviour of the k−regular sets in the quotient space X/G. (see Theorem 6.3 [17]). We will also make one which we will refer to as the isotropy condition I.…”

“…Amongst other results they obtained the following: Theorem 2.20. (Quotients [17]). Let (X, d, m) be an RCD * (K, N ) space and G ≤ Iso(X) a compact subgroup of measure preserving isometries.…”

“…Theorem 2.21. (Principal orbit theorem [17] ) Let (X, d, m) be an RCD * (K, N ) space and G ≤ Iso(X) a compact group of measure preserving isometries. Fix an orbit G · x ∈ X/G.…”

“…As mentioned before, in [17] an additional condition (see Definition 5.4 ) was assumed in order to obtain information on the dimension of the tangents spaces of X/G. Here we define a weaker assumption which will be suficient for us.…”