  On a compact $n$-dimensional manifold $M$, it is well known that a critical
metric of the total scalar curvature, restricted to the space of metrics with
unit volume, is Einstein. It has been conjectured that a critical metric of the
total scalar curvature, restricted to the space of metrics with constant scalar
curvature of unit volume, will also be Einstein. It was shown that this
conjecture is true when $M$ together with a critical metric has harmonic
curvature or the metric is Bach flat. In this paper, we tried to prove this
conjecture with a divergence-free Bach tensor.
