Manifolds with harmonic Weyl curvature and curvature operator of the second kind

TL;DR

This paper classifies certain compact Riemannian manifolds with harmonic Weyl curvature and specific curvature operator conditions, showing they are either conformally equivalent to constant curvature spaces or flat, extending previous classifications.

Contribution

It generalizes existing results by classifying higher-dimensional manifolds with harmonic Weyl curvature under curvature operator constraints.

Findings

Manifolds are either conformally equivalent to positive constant curvature spaces or flat.

Provides a classification for four-dimensional manifolds with harmonic Weyl curvature under cone conditions.

Extends previous work to higher dimensions and broader curvature conditions.

Abstract

We prove that a compact Riemannian manifold of dimension $n\ge 8$ with harmonic Weyl curvature and $\frac{3(n-1)(n+2)}{4(3n-1)}$-nonnegative curvature operator of the second kind is either globally conformally equivalent to a space of positive constant curvature or is isometric to a flat manifold. In particular, We also give a classification of four-dimensional manifolds with harmonic Weyl curvature satisfying a cone condition. This result generalizes the work in \cite{DFY24,FLD,Li22}.