A new approach to the study of spacelike submanifolds in a spherical Robertson-Walker spacetime: characterization of the stationary spacelike submanifolds as an application

Abstract

A natural one codimension isometric embedding of each $(n+1)$-dimensional spherical Robertson-Walker (RW) spacetime $I{\times }_f{\mathbb{S}}^n$ in $(n+2)$-dimensional Lorentz-Minkowski spacetime ${\mathbb{L}}^{n+2}$ permits to contemplate $I{\times }_f{\mathbb{S}}^n$ as a rotation Lorentzian hypersurface of ${\mathbb{L}}^{n+2}$. After a detailed study of such Lorentzian hypersurfaces, any $k$-dimensional spacelike submanifold of such an RW spacetime can be contemplated as a spacelike submanifold of ${\mathbb{L}}^{n+2}$. Then, we use that situation to study $k$-dimensional stationary (i.e., of zero mean curvature vector field) spacelike submanifolds of the RW spacetime. In particular, we prove a wide extension of the Lorentzian version of the classical Takahashi theorem, giving a characterization of stationary spacelike submanifolds of $I{\times }_f{\mathbb{S}}^n$ when contemplating them as spacelike submanifolds of ${\mathbb{L}}^{n+2}$.