Über die Isometriegruppe bei kompakten Mannigfaltigkeiten negative Krümmung

Hof, Hans-Christoph Im

doi:10.1007/bf02566108

Cited by 14 publications

(4 citation statements)

References 6 publications

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“…This result already shows there is an interesting connection between the question of bounds on the order of Isom(M, g) and the topology of M . Further results in this direction have been proved by Huber [Hub71] (for hyperbolic manifolds), Im Hof (for negatively curved manifolds) [Hof73], Maeda (for nonpositively curved manifolds with negative Ricci curvature) [Mae75], and Katsuda (for manifolds with negative Ricci curvature) [Kat88], and DaiShen-Wei [DSW94]. For more information see [DSW94] and the references therein.…”

Section: Introductionmentioning

confidence: 90%

Symmetry gaps in Riemannian geometry and minimal orbifolds

Limbeek

2017

J. Differential Geom.

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Abstract. We study the size of the isometry group Isom(M, g) of Riemannian manifolds (M, g) as g varies. For M not admitting a circle action, we show that the order of Isom(M, g) can be universally bounded in terms of the bounds on Ricci curvature, diameter, and injectivity radius of M . This generalizes results known for negative Ricci curvature to all manifolds.More generally we establish a similar universal bound on the index of the deck group π1(M ) in the isometry group Isom( M , g) of the universal cover M in the absence of suitable actions by connected groups. We apply this to characterize locally symmetric spaces by their symmetry in covers. This proves a conjecture of Farb and Weinberger with the additional assumption of bounds on curvature, diameter, and injectivity radius. Further we generalize results of Kazhdan-Margulis and Gromov on minimal orbifolds of nonpositively curved manifolds to arbitrary manifolds with only a purely topological assumption.

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Section: Introductionmentioning

confidence: 90%

Symmetry gaps in Riemannian geometry and minimal orbifolds

Limbeek

2017

J. Differential Geom.

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“…This result already shows there is an interesting connection between the question of bounds on the order of Isom(M, g) and the topology of M . Further results in this direction have been proved by Huber [Hub71] (for hyperbolic manifolds), Im Hof (for negatively curved manifolds) [Hof73], Maeda (for nonpositively curved manifolds with negative Ricci curvature) [Mae75], and Katsuda (for manifolds with negative Ricci curvature) [Kat88], and Dai-Shen-Wei [DSW94]. For more information see [DSW94] and the references therein.…”

Section: Introductionmentioning

confidence: 90%

Symmetry gaps in Riemannian geometry and minimal orbifolds

Limbeek¹

2014

Preprint

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We study the size of the isometry group Isom(M, g) of Riemannian manifolds (M, g) as g varies. For M not admitting a circle action, we show that the order of Isom(M, g) can be universally bounded in terms of the bounds on Ricci curvature, diameter, and injectivity radius of M . This generalizes results known for negative Ricci curvature to all manifolds.More generally we establish a similar universal bound on the index of the deck group π1(M ) in the isometry group Isom( M , g) of the universal cover M in the absence of suitable actions by connected groups. We apply this to characterize locally symmetric spaces by their symmetry in covers. This proves a conjecture of Farb and Weinberger with the additional assumption of bounds on curvature, diameter, and injectivity radius. Further we generalize results of Kazhdan-Margulis and Gromov on minimal orbifolds of nonpositively curved manifolds to arbitrary manifolds with only a purely topological assumption.

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“…Fifty years before Bochner's result came out, Hurwitz [19] showed that when X is a Riemann surface of genus g ≥ 2, the order of the automorphism group of X, # Aut(X) ≤ 84(g − 1). Later, the estimate of the order of the isometry group was generalized to hyperbolic manifolds by Huber [18], to manifolds with sectional curvature bounded above from 0 by Im Hof [20], to manifolds with non-positive sectional curvature and Ricci curvature negative at some point by Maeda [25] and to manifolds with non-positive sectional curvature and finite volume by Yamaguchi [34]. For general compact Riemannian manifolds with negative Ricci curvature, Katsuda [21] estimated the order of the isometry group by sectional curvature, dimension, diameter and injectivity radius.…”

Section: Bochner's Theoremmentioning

confidence: 99%

The Measure Preserving Isometry Groups of Metric Measure Spaces

Guo¹

2020

SIGMA

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Bochner's theorem says that if M is a compact Riemannian manifold with negative Ricci curvature, then the isometry group Iso(M) is finite. In this article, we show that if (X,d,m) is a compact metric measure space with synthetic negative Ricci curvature in Sturm's sense, then the measure preserving isometry group Iso(X,d,m) is finite. We also give an effective estimate on the order of the measure preserving isometry group for a compact weighted Riemannian manifold with negative Bakry-Emery Ricci curvature except for small portions.

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Über die Isometriegruppe bei kompakten Mannigfaltigkeiten negative Krümmung

Cited by 14 publications

References 6 publications

Symmetry gaps in Riemannian geometry and minimal orbifolds

Symmetry gaps in Riemannian geometry and minimal orbifolds

Symmetry gaps in Riemannian geometry and minimal orbifolds

The Measure Preserving Isometry Groups of Metric Measure Spaces

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