Noncompact complete manifolds with cyclic parallel Ricci curvature

Yawei Chu¹

¹ School 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

Annales Polonici Mathematici, Tome 119 (2017) no. 2, pp. 95-105.

Résumé

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.

Détail

DOI : 10.4064/ap4123-3-2017

Keywords: noncompact complete n dimensional riemannian manifold cyclic parallel ricci curvature positive yamabe constant scalar curvature negative assuming beta norms see theorem range nbsp beta weyl curvature finite space form norms traceless ricci curvature weyl curvature small enough rigidity result obtained dimensions without assumption beta norms weyl curvature