Some rigidity results on shrinking gradient Ricci soliton

Abstract

Suppose $(M^n,g,f)$ is a complete shrinking gradient Ricci soliton. We give several rigidity results under some natural conditions, generalizing the results in \cite{Petersen-Wylie,Guan-Lu-Xu}. Using maximum principle, we prove that shrinking gradient Ricci soliton with constant scalar curvature $R=1$ is isometric to a finite quotient of ${\mathbb{R}}^2\times {\mathbb{S}}^2$, giving a new proof of the main results of Cheng-Zhou \cite{Cheng-Zhou}.