Rigidity theorem for complete spacelike submanifold in \(S^{n+p}_q(1)\) with constant scalar curvature
The main theorem is as follows: Let \(M\) be a complete space-like \(n\)-submanifold of a Lorentzian space form of curvature 1. Assume that \(M\) has parallel mean curvature and constant scalar curvature \(\leq 1\). Assume furthermore that the squared norm of the second fundamental form of \(M\) is bounded from above by \(2\sqrt{n-1}\). Then \(M\) is either totally umbilical or isometric to a hyperbolic cylinder. The proof uses Cheng-Yau's technique from [\textit{S.-Y. Cheng} and \textit{S.-T. Yau}, Ann. Math. (2) 104, 407--419 (1976; Zbl 0352.53021)].
scientific article; zbMATH DE number 6126487
scientific article