On compact Riemannian manifolds with harmonic weyl curvature

TL;DR

This paper establishes rigidity results for compact Riemannian manifolds with harmonic Weyl curvature, positive scalar curvature, and positive constant , including a classification of certain 4-dimensional cases as quotients of the sphere.

Contribution

It provides new rigidity theorems for manifolds with harmonic Weyl curvature and characterizes 4-dimensional locally conformally flat cases as quotients of the sphere.

Findings

Rigidity theorems for manifolds with harmonic Weyl curvature

Classification of 4D locally conformally flat manifolds with positive scalar curvature

Identification of manifolds as quotients of the round sphere

Abstract

We give some rigidity theorems for an n$(\geq4)$-dimensional compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant $\sigma_2$. Moreover, when $n=4,$ we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positive constant $\sigma_2$ is isometric to a quotient of the round $\mathbb{S}^4$.