Rigidity theorems for complete spacelike submanifold in \(S^{n+p}_q(1)\) (Q2248884)

The paper under review concerns space-like submanifolds, whose mean curvature \(H\) and normalized scalar curvature \(R\) satisfy a linear relation \(R=aH+b\) with constant coefficients \(a\) and \(b\), in a (\(n+p\))-dimensional semi-Riemannian manifold \(\mathbb S^{n+p}_q(1)\) of index \(q\) and of sectional curvature \(1\). The following results characterizing totally umbilical submanifolds in \(\mathbb S^{n+p}_q(1)\) are proved, cf. also [\textit{F. E. C. Camargo} et al., Differ. Geom. Appl. 26, No. 6, 592--599 (2008; Zbl 1160.53361); Kodai Math. J. 32, No. 2, 209--230 (2009; Zbl 1171.53032); \textit{Y. Han} and \textit{S. Feng}, Bull. Math. Anal. Appl. 3, 97--108 (2011); \textit{R. M. B. Chaves} and \textit{L. A. M. Sousa jun.}, Differ. Geom. Appl. 25, No. 4, 419--432 (2007; Zbl 1143.53059)]: Theorem 1. Let \(x: M^n \to \mathbb S^{n+p}_q(1)\), \(n\geq 3\), \(1\leq q\leq p\), be a substantial isometric immersion of a complete Riemannian manifold. Assume that the normalized scalar curvature \(R\) of \(M^n\) in \(\mathbb S^{n+p}_q(1)\) satisfies \(R=aH+b\), \((n-1)a^2+4n(1-b)\geq 0\), \(a\geq 0\). If the squared norm \(S\) of the second fundamental form satisfies \(\sup S\leq 2\sqrt{n-1}\), then either (1) \(S=nH^2\), \(M^n\) is a totally umbilical submanifold and \(S=n(1-R)\), or (2) \(\sup S= 2\sqrt{n-1}\) and \(M^n\), if \(n=2\), is a totally umbilical submanifold, or if \(M^n\), \(n\geq 3\), lies in a totally geodesic submanifold \(\mathbb S^{n+1}_1(1)\subset \mathbb S^{n+p}_q(1)\) and \(M^n\) is isometric to a hyperbolic cylinder \(\mathbb S^{n-1}(1-\tanh^2r)\times \mathbb H^{1}(1-\coth^2r)\). Theorem 2. Let \(M^n\) be a complete space-like submanifold with \(R=aH+b\), \((n-1)a^2+4n(1-b)\geq 0\), \(a\geq 0\), in \(\mathbb S^{n+p}_q(1)\), \(1\leq q\leq p\). Let \(\sup K\) be the function that assigns to each point \(p\in M^n\) the supremum of the sectional curvature of \(M^n\) at \(p\). There exists a constant \(\beta (n,q,H)\) such that if \(\sup K\leq \beta (n,q,H)\) then either (1) \(n=2\) and \(M^n\) is totally umbilical, or (2) \(n\geq 3\) and \(M^n\) is totally geodesic.