Rigidity of Riemannian manifolds with positive scalar curvature

TL;DR

This paper establishes rigidity results for Bach-flat closed Riemannian manifolds with positive scalar curvature, linking curvature inequalities to geometric rigidity, especially in four dimensions.

Contribution

It introduces new rigidity theorems under curvature inequalities involving Weyl and traceless Ricci curvatures, extending previous results to broader classes of manifolds.

Findings

Rigidity results for Bach-flat manifolds with positive scalar curvature

New inequalities involving Weyl curvature and traceless Ricci curvature

Applications to rigidity in 4-dimensional manifolds

Abstract

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity result under a given inequality involving the Weyl curvature and the traceless Ricci curvature. Moveover, under an inequality involving $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and the Yamabe invariant, we also provide a similar rigidity result. As an application, we obtain some rigidity results on 4-dimensional manifolds.