  On a compact n-dimensional manifold, it has been conjectured that a critical
point metric of the total scalar curvature, restricted to the space of metrics
with constant scalar curvature of unit volume, will be Einstein. This
conjecture was proposed in 1984 by Besse, but has yet to be proved. In this
paper, we prove that if the manifold with the critical point metric has
harmonic curvature, then it is isometric to a standard sphere.
