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Noncompact complete manifolds with cyclic parallel Ricci curvature

Volume 119 / 2017

Yawei Chu Annales Polonici Mathematici 119 (2017), 95-105 MSC: Primary 53C24; Secondary 53C20. DOI: 10.4064/ap4123-3-2017 Published online: 11 July 2017

Abstract

Let $(M^n,g)$ be a noncompact complete $n$-dimensional Riemannian manifold with cyclic parallel Ricci curvature and positive Yamabe constant. When the scalar curvature $R$ is negative, assuming that the $L^\beta $-norms (see Theorem 1.1 for the range of $\beta $) of the Weyl curvature are finite, we show that $(M^n,g)$ is a space form if $n\ge 7$ and the $L^{n/2}$-norms of the traceless Ricci curvature and Weyl curvature are small enough. When $R=0,$ the same rigidity result is also obtained for all dimensions $n\ge 3$ without the assumption on the $L^\beta $-norms of the Weyl curvature.

Authors

  • Yawei ChuSchool of Mathematics and Statistics
    Fuyang Normal University
    Fuyang, 236037, People’s Republic of China
    and
    College of Information Engineering
    Fuyang Normal University
    Fuyang, 236041, People’s Republic of China
    e-mail

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