Space-like hypersurfaces with constant scalar curvature in the de Sitter spaces (Q1917365)

The author proves the following Theorem: Let \(M^n\) be an \(n\)-dimensional compact space-like hypersurface in \(M^{n+1}_1 (c)\) with constant scalar curvature. If \(M^n\) satisfies \(K(M) > 0\), \(\text{Ric} (M) \leq (n -1)c\), \(R \leq c\), where \(R\) is the normalized scalar curvature of \(M^n\), then \(M^n\) is totally umbilical.