  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.
