The Isometry Group of an RCD ∗ Space is Lie

Sosa, Gerardo

doi:10.1007/s11118-017-9656-4

Cited by 15 publications

(21 citation statements)

References 34 publications

(76 reference statements)

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“…The following lemma was observed by the fourth author [68] and, independently, by Guijarro-Santos-Rodríguez [38]. We present an independent argument for the reader's convenience.…”

Section: 2mentioning

confidence: 62%

“…y ∈ M , it holds that the isotropy groups G y = gG x0 g −1 and G x0 are conjugate. Thus, since the action of G is effective and for every x ∈ M , G/G x acts transitively on the orbit G(x) we conclude that the orbits [46] by Ketterer and Rajala, and generalized in [68] by the fourth author. We note that none of these spaces has good transport behavior, or equivalently in this case, the essential non-branching condition.…”

Section: Principal Orbit Theorem and Cohomogeneity One Actionsmentioning

confidence: 73%

“…We denote by M * := M/G the quotient space and by p : M → M * the quotient map. Furthermore, we endow M * with the pushforward measure m * := p ♯ m and the quotient metric It has recently been shown by Guijarro and Santos-Rodríguez [38], and independently by the the fourth author [68], that the group of (measure-preserving) isometries of an RCD * (K, N ) space is a Lie group. Moreover, as shown in [68], the corresponding statements are also valid for strong CD/CD * (K, N )-spaces and essentially non-branching MCP-spaces for which tangent cones are well-behaved, yet might fail to be…”

Section: Introductionmentioning

confidence: 99%

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On quotients of spaces with Ricci curvature bounded below

Galaz-García

Kell

Mondino

et al. 2018

Journal of Functional Analysis

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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.

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“…The following lemma was observed by the fourth author [68] and, independently, by Guijarro-Santos-Rodríguez [38]. We present an independent argument for the reader's convenience.…”

Section: 2mentioning

confidence: 62%

Section: Principal Orbit Theorem and Cohomogeneity One Actionsmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 99%

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On quotients of spaces with Ricci curvature bounded below

Galaz-García

Kell

Mondino

et al. 2018

Journal of Functional Analysis

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“…In [24], and indenpendently in [37], it was shown that the isometry group of an RCD * (K, N ) space is a Lie group. Therefore it makes sense now to study properties of Lie group actions on m.m.s.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…A priori one would assume that the metric structure and the measure have no relationship at all; however we have the following result. The following was obtained in [24], and independently by Sosa [37]. Theorem 2.19.…”

Section: 5mentioning

confidence: 90%

Invariant Measures and Lower Ricci Curvature Bounds

Santos-Rodríguez

2019

Potential Anal

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Given a metric measure space (X, d, m) that satisfies the Riemannian Curvature Dimension condition, RCD * (K, N ), and a compact subgroup of isometries G ≤ Iso(X) we prove that there exists a G−invariant measure, mG, equivalent to m such that (X, d, mG) is still a RCD * (K, N ) space. We also obtain some applications to Lie group actions on RCD * (K, N ) spaces. We look at homogeneous spaces, symmetric spaces and obtain dimensional gaps for closed subgroups of isometries.

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