  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.
