Negative Ricci curvature and isometry group
type: Item
instance of: scholarly article
title: Negative Ricci curvature and isometry group
zbMATH Open document ID: 0820.53045
DOI: 10.1215/S0012-7094-94-07603-5
author: Xianzhe Dai
author: Guofang Wei
author: Zhongmin Shen
published in: Duke Mathematical Journal
publication date: 1995-09-05T00:00:00Z
review text: Given positive constants \(\lambda\), \(\Lambda\), \(i_ 0\), \(V\) and a positive integer \(n\) let \({\mathcal M} = {\mathcal M}(n, \lambda, \Lambda, i_ 0, V)\) denote the class of Riemannian \(n\)-manifolds \(M\) satisfying \(- \Lambda \leq \text{Ric} \leq -\lambda < 0\), \(\text{inj} \geq i_ 0 > 0\) and \(\text{vol} \leq V\), where Ric, inj, and vol denote the Ricci curvature, injectivity radius and volume of \(M\) respectively. M. Anderson has shown that \(\mathcal M\) is precompact in the \(C^{1,\alpha}\) topology and has obtained uniform \(L^ p\) bounds for the curvature tensor of manifolds in \(\mathcal M\). Let \(I(M)\) and \(| I(M)|\) denote the isometry group of \(M\) and its cardinality, respectively. In this article the authors prove the following Theorem A. Let \(\{M_ i\}\) be a sequence of manifolds in \({\mathcal M}(n, \lambda, \Lambda, i_ 0, V)\) that converges to a \(C^{1,\alpha}\) Riemannian manifold \(M\). Then (a) \(| I(M)|\) is finite, and (b) \(\lim_{i \to \infty} \sup | I(M_ i)| \leq | I(M)|\). From this result one obtains the following Corollary. There exists a positive integer \(N = N(n, \lambda, \Lambda, i_ 0, V)\) such that \(| I(M)| \leq N\) for every \(M \in {\mathcal M}(n, \lambda, \Lambda, i_ 0, V)\). Katsuda has previously proved the corollary under the additional hypothesis that the sectional curvatures of manifolds in \({\mathcal M}(n,\lambda, \Lambda, i_ 0, V)\) are uniformly bounded below. By a result of T. Yamaguchi one can drop the lower bound on the injectivity radius if one assumes that the sectional curvature satisfies \(-\Lambda \leq K < 0\). The theorem above is a consequence of the next result on the displacement function \(d_ \varphi : x \to d(x,\varphi x)\) of an isometry \(\varphi\). Theorem B. Let \(M \in {\mathcal M}(n, \lambda, \Lambda, i_ 0, V)\). Then there is a constant \(\varepsilon_ 0 = \varepsilon_ 0 (n, \lambda, \Lambda, i_ 0, V)\) such that if \(\varphi\) is an isometry of \(M\) satisfying \(d_ \varphi(x) \le…
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 727819
zbMATH Keywords: negative Ricci curvature
zbMATH Keywords: finite isometry group
zbMATH Keywords: bounded geometry
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
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full work available at URL: s0012-7094-94-07603-5
OpenAlex ID: W2143154157
description: scientific article; zbMATH DE number 727819