Ricci deformation of the metric on Riemannian orbifolds
type: Item
instance of: scholarly article
title: Ricci deformation of the metric on Riemannian orbifolds
zbMATH Open document ID: 1203.53066
author: Jia-Yong Wu
published in: Results in Mathematics
publication date: 2010-06-23T00:00:00Z
review text: The author extends to dimensions \(n\geq 4\) a result due to Hamilton [Three-orbifolds with positive Ricci curvature, pp. 521--524 in \textit{H. D. Cao} et al., Collected Papers on Ricci Flow. Series in Geometry and Topology 37. Somerville, MA: International Press (2003; Zbl 1108.53002)] that a compact three-orbifold which admits a Riemannian metric of positive Ricci curvature also admits a metric of constant positive sectional curvature, so that the manifold is diffeomorphic to the quotient of the sphere \(S^{3}\) by a finite group of isometries. This then extends to orbifolds a result due to \textit{G. Huiskin} [J. Differ. Geom. No.~21, 47--62 (1985; Zbl 0606.53026)] that a compact Riemannian manifold of dimension \(n\geq 4\) of positive scalar curvature admits a metric of constant positive sectional curvature under the assumption that in the decomposition of the Riemannian curvature tensor \(R=W+V+U,\) the norms of the Weyl conformal curvature tensor \(W\) and the traceless part of the Ricci tensor \(V\) are pointwise not large compared with the scalar curvature part \(U\). In particular the author proves that on a compact Riemannian orbifold \((M,g)\) of dimension \(n\geq 4\) if the curvature tensor with positive scalar curvature satisfies the condition that \(\left| W\right| ^{2}+\left| V\right| ^{2}<\delta _{n}\left| U\right| ^{2}\) with \(\delta _{5}=\frac{1}{10}\) and \(\delta _{n}=\frac{2}{\left( n-2\right) \left( n+1\right) }\) for \(n\neq 5\) then \(M\) admits a metric of constant positive curvature. The proof uses the maximum principle for parabolic equations.
reviewed by: Lew Friedland
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 5725195
zbMATH Keywords: Ricci flow
zbMATH Keywords: orbifold
zbMATH Keywords: metric deformation
zbMATH Keywords: constant curvature
Wikidata QID: Q115389906
MaRDI profile type: Publication
full work available at URL: s00025-010-0036-2
OpenAlex ID: W2043491567
cites work: Manifolds with positive curvature operators are space forms
cites work: Q4298656
cites work: Orbifolds with lower Ricci curvature bounds
cites work: The splitting theorem for orbifolds
cites work: On the entropy estimate for the Ricci flow on compact 2-orbifolds
cites work: The Ricci flow on compact 2‐orbifolds with curvature negative somewhere
cites work: Deforming metrics in the direction of their Ricci tensors
cites work: Three-manifolds with positive Ricci curvature
cites work: Ricci deformation of the metric on a Riemannian manifold
cites work: A compactness property for solutions of the Ricci flow on orbifolds
DOI: 10.1007/S00025-010-0036-2
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description: scientific article; zbMATH DE number 5725195