Rigidity of complete generic shrinking Ricci solitons

Let (Mn,g,X) be a complete generic shrinking Ricci soliton of dimension n≥3. In this paper, by employing curvature inequalities, the formula of X-Laplacian for the norm square of the trace-free curvature tensor, the weak maximum principle and the estimate of the scalar curvature of (Mn,g), we prove some rigidity results for (Mn,g,X). In particular, it is showed that (Mn,g,X) is isometric to Rn or a finite quotient of Sn under a pointwise pinching condition. Moreover, we establish several optimal inequalities and classify those shrinking solitons for equalities.