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}.
Categories: math.DG