Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

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.