Space-like submanifolds with constant scalar curvature (Q5949960)

Let \(M^n\) be a space-like submanifold in a de Sitter space \(M_p^{n+p}(c)\) with constant scalar curvature. This letter gives intrinsic conditions for \(M^n\) to be totally umbilical. The main result is as follows: Let \(M^n\) be an \(n\)-dimensional compact space-like submanifold with constant scalar curvature and with nonnegative sectional curvature immersed in \(M_p^{n+p}(c)\). Suppose that \(M^n\) has a flat normal bundle. If the normalized scalar curvature \(R\) of \(M^n\) satisfies \(R<c\), then \(M^n\) is totally umbilical and isometric to a sphere.