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

TL;DR

This paper characterizes stationary spacelike submanifolds in spherical Robertson-Walker spacetimes by embedding them into Lorentz-Minkowski space and extending classical theorems to this setting.

Contribution

It introduces a new approach to study spacelike submanifolds in RW spacetimes via Lorentzian hypersurface embeddings and extends the Takahashi theorem to this context.

Findings

Characterization of stationary spacelike submanifolds in RW spacetimes.

Extension of the Lorentzian Takahashi theorem.

Embedding of RW spacetimes as rotation hypersurfaces in Lorentz-Minkowski space.

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 $\mathbb{L}^{n+2}$.