On isometry of a complete Riemannian manifold to a sphere
The purpose of this article is to obtain sufficient conditions in terms of conformal vector fields and curvatures for a complete Riemannian manifold to be isometric to an ordinary \(n\)-sphere. Among others, he proves the following theorem. Let \(M\) be a complete Riemannian manifold of dimension \(>2\) with non-zero constant scalar curvature. Let \(R\) and \(K\) denote the Riemannian curvature tensor and the Ricci tensor of \(M\), respectively. If \(M\) admits a non-isometric conformal vector field \(X\), \(\mathcal L_ Xg = 2\rho g\), such that \({\mathcal L}_ X| R|^ 2 = 0\) (or \({\mathcal L}_ X| K|^ 2 = 0\)) and if the \(\rho\) has first derivative in \(L^ 2(M)\), then \(M\) is isometric to an ordinary \(n\)- sphere.
scientific article; zbMATH DE number 701553