Bulletin of the
Korean Mathematical Society BKMS

On a compact $n$-dimensional manifold $M$, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, after derivng an interesting curvature identity, we show that the conjecture is true in dimension three and four when $g$ is weakly Einstein. In higher dimensional case $n\geq 5$, we also show that the conjecture is true under an additional Ricci curvature bound. Moreover, we prove that the manifold is isometric to a standard $n$-sphere when it is $n$-dimensional weakly Einstein and the kernel of the linearized scalar curvature operator is nontrivial.

Keywords: total scalar curvature, critical point metric, weakly Einstein, Einstein metric, linearized scalar curvature

MSC numbers: Primary 53C25, 58E11