Compact manifolds of constant scalar curvature (Q1176341)

It is an interesting problem to characterize the collection of Riemannian manifolds which are isometric to a sphere in the class of compact non- negatively curved Riemannian manifolds of constant scalar curvature. Concerning this problem, \textit{A. Ros} has proved that a compact Riemannian manifold of constant scalar curvature which admits an isometric embedding in Euclidean space as a hypersurface is isometric to a sphere [J. Differ. Geom. 27, No. 2, 215-220 (1988; Zbl 0638.53051)]. The authors prove that if an \(n\)-dimensional compact, connected and non-negatively curved Riemannian manifold \(M\) of constant scalar curvature satisfies the conditions, \((i)\;(\nabla_ x\hbox{Ric})(Y,Z)=(\nabla_ Y\hbox{Ric})(X,Z)\), \(X,Y,Z\in{\mathcal X}(M)\), and (ii) \(I^ 0(M)\neq C^ 0(M)\), then \(M\) is isometric to an \(n\)- dimensional sphere \(S^ n\), where \(I^ 0(M)\) (resp. \(C^ 0(M)\)) denotes the identity component of the isometry group (resp. the conformal transformation group) of \(M\). The authors prove this result by making use of an integral formula established by \textit{A. Ros} [Ann. Math., II. Ser. 121, 377-382 (1985; Zbl 0589.53065)].