The isometry groups of compact manifolds with almost negative Ricci curvature
type: Item
instance of: scholarly article
title: The isometry groups of compact manifolds with almost negative Ricci curvature
zbMATH Open document ID: 1406.53042
author: Atsushi Katsuda
author: Takeshi Kobayashi
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2018-12-21T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tmj/1537495353
review text: Let \(M\) be a compact Riemannian manifold of dimension \(n\geq3\). Bochner showed that if the Ricci tensor of \(M\) is negative, then there are no non-trivial Killing vector fields and thus the isometry group \(I(M)\) is finite. The authors generalize this result to assume that \(M\) has negative Ricci curvature except in small portions. To make this precise, consider the Sobolev inequality \[ \|f\|^2_{\frac{2n}{n-2}}\leq A\|\nabla f\|^2_2+B\|f\|^2_2\,. \] Let \(\varrho(P):=\max_{X\in T_PM,\|X\|=1}\mathrm{Ric}(X,X)\). Let \((\varrho+\omega)_+=\max(0,\varrho+\omega)\) where \(\omega\) is a positive constant. Theorem. Suppose there exists \(\omega>0\) so that \(\|(\varrho+\omega)_+\|_{\frac n2}<\min\left(\frac1A,\frac\omega B\right)\). Then \(M\) has no non-trivial Killing vector fields and consequently \(I(M)\) is finite. If one assumes that \(\|(\rho+\omega)_+\|<\min\left(\frac1{2A},\frac\omega{2B}\right)\), the authors discuss bounds for the isometry group in terms of various geometric quantities.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 6996534
zbMATH Keywords: isometry group
zbMATH Keywords: comparison theorem
MaRDI profile type: Publication
OpenAlex ID: W2891358872
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DOI: 10.2748/TMJ/1537495353
description: scientific article; zbMATH DE number 6996534