Bulletin of the
Korean Mathematical Society BKMS

It has been conjectured that, on a compact orientable manifold $M$, a critical point of the total scalar curvature functional restricted the space of unit volume metrics of constant scalar curvature is Einstein. In this paper we show that if a manifold is a $3$-dimensional warped product, then $(M,g)$ cannot be a critical point unless it is isometric to the standard sphere.

Keywords: total scalar curvature functional, critical point equation, Einstein metric

MSC numbers: 52C25