On Chen surfaces in a Minkowski space time (Q1106486)

Let \(E_ s^{m+1}\) denote the flat \((m+1)\)-dimensional pseudo-Euclidean space of signature \((s,m+1-s)\). Denote by \(S^ m_ s\) and \(H^ m_{s- 1}\) the standard pseudo-Riemannian sphere and the pseudo-hyperbolic space in \(E_ s^{m+1}\), respectively. Among others, the following results are obtained. Theorem 1. Let M be a (pseudo-Riemannian) submanifold of \(S^ m_ s\) (or \(H^ m_{s-1})\) in \(E_ s^{m+1}\). If M is a Chen submanifold of \(E_ s^{m+1}\), then M is a Chen submanifold of \(S_ s^ m\) (or of \(H^ m_{s-1})\). Theorem 2. Let M be a spacelike surface in \(S^ 3_ 1\) (or in \(H^ 3_ 0)\) in the Minkowski space-time \(E^ 4_ 1\). Then M is minimal in \(S^ 3_ 1\) (or in \(H^ 3_ 0)\) if and only if M is a Chen surface. Theorem 3. Let M be a pseudo-umbilical surface of signature (1,1) in \(S^ 3_ 1\). If the Weingarten map in the direction of the mean curvature vector has a double eigenvalue, then M is a Chen surface. If it has two different eigenvalues and M is a Chen surface, then M is minimal in \(S^ 3_ 1\).