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 [14, 25]. 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 [9].
