Riemannian manifolds with harmonic curvature
Volume 145 / 2016
Colloquium Mathematicum 145 (2016), 251-257
MSC: Primary 53C20; Secondary 53C21.
DOI: 10.4064/cm6826-4-2016
Published online: 4 July 2016
Abstract
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.
