A Simons Type Formula for Spacelike Submanifolds in Semi-Riemannian Warped Product and its Applications

We determine a Simons type formula for spacelike submanifolds within a broad class of semi-Riemannian warped products, which includes the Robertson-Walker spacetimes. This formula enables us to extend results well-known from Riemannian geometry to the semi-Riemannian case. Specifically, when the ambient space has constant sectional curvature, such as Lorentz-Minkowski, de Sitter and anti-de Sitter spacetimes, we establish that compact spacelike 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 by Nomizu and Smyth in 1969. As an application, we extend analogous results previously established for pseudo-parallel immersions in Riemannian space forms. With this extension, we prove that any semi-parallel spacelike hypersurface with zero mean curvature into de Sitter spacetime is totally geodesic. Furthermore, we show that there are no semi-parallel spacelike hypersurfaces with zero mean curvature into Einstein-de Sitter spacetime.