A note on compact hypersurfaces in a Euclidean space
An affirmative answer to the question ``Does an orientable compact hypersurface in a Euclidean space that admits a unit Killing vector field \(\xi\) together with the condition that \(g(A\xi,\xi)\) is a constant necessary isometric to a sphere?'' is given in the form of the following theorem. Theorem. Let \(M\) be a compact orientable hypersurface of the Euclidean space \(\mathbb R^{n+1}\). \(M\) admits a unit Killing vector field \(\xi\) satisfying that, with respect to the induced metric \(g\) and the shape operator \(A\), \(g(A\xi,\xi)\) is a constant if and only if \(n=2m+1\) and \(M\) is isometric to the sphere \(S^{2m+1}(c)\) of constant curvature \(c\).
scientific article; zbMATH DE number 6115455