Remarks on compact shrinking Ricci solitons of dimension four (Q387783)

A gradient shrinking Ricci soliton \((M,g)\) is defined by the equation \(\text{Ric}=\rho g+ D^2f\) where Ric is the Ricci tensor, \(\rho> 0\) is a real constant and \(D^2f\) is the Hessian of a smooth function \(f\). For a compact 4-dimensional gradient shrinking Ricci soliton, the author obtains the equation NEWLINE\[NEWLINE\int_M SdV= 4\rho\text{\,Vol}(M)NEWLINE\]NEWLINE where \(S\) is the scalar curvature.NEWLINENEWLINE In addition, two results on topological constants of the manifold are proved. These involve inequalities between the Euler characteristic \(\chi\) and the signature \(\tau\). One of them coincides with the Hitchin-Thorpe inequality \(|\tau|\leq{2\over 3}\chi\) for compact 4-dimensional Einstein spaces.