Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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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.

Critical point metrics of the total scalar curvature

Jeongwook Chang, Seungsu Hwang, and Gabjin Yun

Dankook University, Chung-Ang University, Myong Ji University

Abstract

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