Some rigidity characterizations on critical metrics for quadratic curvature functionals

TL;DR

This paper investigates critical metrics for quadratic curvature functionals on closed manifolds, providing classifications and conditions under which these metrics are Einstein, thus advancing understanding of geometric structures related to curvature functionals.

Contribution

It offers new classifications of critical metrics for quadratic curvature functionals and establishes conditions under which these metrics are necessarily Einstein.

Findings

Classification of critical metrics under integral conditions

Proof that critical metrics are Einstein under certain curvature conditions

Extension of rigidity results for quadratic curvature functionals

Abstract

We study closed $n$-dimensional manifolds of which the metrics are critical for quadratic curvature functionals involving the Ricci curvature, the scalar curvature and the Riemannian curvature tensor on the space of Riemannian metrics with unit volume. Under some additional integral conditions, we classify such manifolds. Moreover, under some curvature conditions, the result that a critical metric must be Einstein is proved.