On some locally symmetric embedded spaces with non-negative scalar curvature and their characterization

TL;DR

This paper studies specific locally symmetric hypersurfaces in spacetimes with a 1+1+2 decomposition, characterizing their geometry, Ricci soliton structures, and conditions for flatness in rotationally symmetric spacetimes.

Contribution

It provides a detailed characterization of locally symmetric hypersurfaces with non-negative scalar curvature, including conditions for Ricci solitons and flatness in spacetimes with local rotational symmetry.

Findings

Hypersurfaces are flat in spacetimes with local rotational symmetry, zero rotation, and spatial twist.

Ricci solitons on these hypersurfaces are necessarily steady with constant soliton components.

Conditions for hypersurfaces to be slices of constant time are explicitly derived.

Abstract

In this work we perform a general study of properties of a class of locally symmetric embedded hypersurfaces in spacetimes admitting a $1+1+2$ spacetime decomposition. The hypersurfaces are given by specifying the form of the Ricci tensor with respect to the induced metric. These are slices of constant time in the spacetime. Firstly, the form of the Ricci tensor for general hypersurfaces is obtained and the conditions under which the general case reduces to those of constant time slices are specified. We provide a characterization of these hypersurfaces, with key physical quantities in the spacetime playing a role in specifying the local geometry of these hypersurfaces. Furthermore, we investigate the case where these hypersurfaces admit a Ricci soliton structure. The particular cases where the vector fields associated to the solitons are Killing or conformal Killing vector fields are analyzed. Finally, in the context of spacetimes with local rotational symmetry it is shown that, only spacetimes in this class with vanishing rotation and spatial twist can admit the hypersurface types considered, and that the hypersurfaces are necessarily flat. And if such hypersurface do admit a Ricci soliton structure, the soliton is steady, with the components of the soliton field being constants.