Riemannian manifolds with harmonic curvature

Guangyue Huang¹ ; Bingqing Ma²

¹College of Mathematics and Information Science Henan Normal University 453007 Xinxiang, P.R. China and Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control
²College of Mathematics and Information Science Henan Normal University 453007 Xinxiang, P.R. China

Colloquium Mathematicum, Tome 145 (2016) no. 2, pp. 251-257

Résumé

We prove an integral inequality for compact $n$-dimensional manifolds with harmonic curvature tensor and positive scalar curvature, generalizing a recent result of Catino that deals with the conformally flat case, and classify those manifolds for which our inequality is an equality: they are either Einstein, $\mathbb {S}^1\times \mathbb {S}^{n-1}$ with the product metric, or $\mathbb {S}^1\times \mathbb {S}^{n-1}$ with a rotationally symmetric Derdziński metric.

Détail

DOI : 10.4064/cm6826-4-2016

Keywords: prove integral inequality compact n dimensional manifolds harmonic curvature tensor positive scalar curvature generalizing recent result catino deals conformally flat classify those manifolds which inequality equality either einstein mathbb times mathbb n product metric mathbb times mathbb n rotationally symmetric derdzi ski metric