A Simons type formula for spacelike submanifolds in semi-Riemannian warped product and its applications (Q6645881)

The authors establish a Simons-type formula for space-like submanifolds in a wide range of semi-Riemannian warped products, including Robertson-Walker spacetimes. This formula enables the extension of results well-known in Riemannian geometry to the semi-Riemannian case. In particular, when the ambient space has constant sectional curvature, such as Lorentz-Minkowski, de Sitter and anti-de Sitter spacetimes, it is proven that compact space-like hypersurfaces with parallel mean curvature vector field and non-negative sectional curvature are isoparametric hypersurfaces. This result constitutes a generalization of the Riemannian case within space forms, as demonstrated in [\textit{K. Nomizu} and \textit{B. Smyth}, J. Differ. Geom. 3, 367--377 (1969; Zbl 0196.25103)]. Furthermore, previous results on pseudo-parallel immersions in Riemannian space forms are extended to semi-parallel space-like hypersurfaces. For instance, it is demonstrated that any semi-parallel space-like hypersurface with zero mean curvature in de Sitter spacetime is totally geodesic. Moreover, it is shown that there are no semi-parallel space-like hypersurfaces with zero mean curvature in Einstein-de Sitter spacetime.