CRITICAL POINT METRICS OF THE TOTAL SCALAR CURVATURE

Abstract

In this paper, we deal with a critical point metric of the total scalar curvature on a compact manifold <TEX>$M$</TEX>. We prove that if the critical point metric has parallel Ricci tensor, then the manifold is isometric to a standard sphere. Moreover, we show that if an <TEX>$n$</TEX>-dimensional Riemannian manifold is a warped product, or has harmonic curvature with non-parallel Ricci tensor, then it cannot be a critical point metric.