Some rigidity characterizations of Einstein metrics as critical points for quadratic curvature functionals

TL;DR

This paper investigates conditions under which Einstein metrics are uniquely characterized as critical points of certain quadratic curvature functionals, providing new rigidity results involving curvature inequalities.

Contribution

It introduces novel rigidity characterizations of Einstein metrics and locally conformally flat metrics via pointwise curvature inequalities.

Findings

Rigidity results for Einstein metrics as critical points of quadratic curvature functionals

Characterizations involving Weyl and traceless Ricci curvature inequalities

Rigidity results for locally conformally flat critical metrics

Abstract

We study rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor, characterized by some pointwise inequalities involving the Weyl curvature and the traceless Ricci curvature. Moreover, we also provide a few rigidity results for locally conformally flat critical metrics.