Ricci deformation of the metric on Riemannian orbifolds
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.
scientific article; zbMATH DE number 5725195