  On a compact $n$-dimensional manifold, 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 be Einstein. This conjecture, proposed in 1987
by Besse, has not been resolved except when $M$ has harmonic curvature or the
metric is Bach flat. In this paper, we prove some gap properties under
divergence-free Bach tensor condition for $n\geq 5$, and a similar condition
for $n=4$.
