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

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