LAUSR

A result showed by M. Gursky in ensures that any metric g on the 4†dimensional sphere S4 satisfying Ricg=3g and injg(S4)≥I€34 is isometric to the round metric. In this note, we prove that there exists a universal number i0 such that any metric g on the 4†dimensional sphere S4 satisfying Ricg=3g and injg(S4)≥I€34−i0 is isometric to the round metric. Moreover, there exists a universal Iµ0>0 such that any metric g on the 4†dimensional sphere S4 with nonnegative sectional curvature, Ricg=3g and 89I€2−Iµ0≤Volg(S4) is isometric to the round metric. This last result slightly improves a rigidity theorem also proved in .