Bull. Korean Math. Soc. 2012; 49(3): 655-667
Printed May 1, 2012
https://doi.org/10.4134/BKMS.2012.49.3.655
Copyright © The Korean Mathematical Society.
Jeongwook Chang, Seungsu Hwang, and Gabjin Yun
Dankook University, Chung-Ang University, Myong Ji University
In this paper, we deal with a critical point metric of the total scalar curvature on a compact manifold $M$. 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 $n$-dimensional Riemannian manifold is a warped product, or has harmonic curvature with non-parallel Ricci tensor, then it cannot be a critical point metric.
Keywords: the total scalar curvature, critical point metric, Einstein
MSC numbers: Primary 53C25
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