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Ricci deformation of the metric on a Riemannian manifold. (English) Zbl 0606.53026
In this paper the heat flow method developed by R. S. Hamilton to deform metrics in the direction of the Ricci curvature is used to prove a local pinching theorem for spherical space forms. The main theorem says that there exists a \(\delta_ n>0\) depending only on \(n>3\) such that if a compact n-dimensional Riemannian manifold M of positive scalar curvature satisfies \(| W|^ 2+| V|^ 2<\delta_ n| U|^ 2\), where W is the Weyl conformal curvature tensor, V is the traceless part of the Ricci tensor and U denotes the scalar curvature part of the Riemannian curvature tensor, then M admits a metric of constant positive sectional curvature. \(\delta_ n\) is given explicitly by \(\delta_ n=2/((n-2)(n+1))\) for \(n\geq 6.\)
The proof is based on a detailed analysis of the different components of the algebraic zeroth order expression appearing in the evolution equation of the whole curvature tensor under Hamilton’s flow. The above result is a significant improvement (using a completely different method) of a theorem of E. A. Ruh, where the constant \(\delta_ n\) could not be computed explicitly.
Reviewer: M.Min-OO

MSC:
53C20 Global Riemannian geometry, including pinching
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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