On some locally symmetric embedded spaces with nonnegative scalar curvature and their characterization
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. First, 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 with 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. If such hypersurface does admit a Ricci soliton structure, the soliton is steady, with the components of the soliton field being constants.
 Publication:

International Journal of Geometric Methods in Modern Physics
 Pub Date:
 March 2022
 DOI:
 10.1142/S0219887822500517
 arXiv:
 arXiv:2112.03219
 Bibcode:
 2022IJGMM..1950051S
 Keywords:

 1 + 1 + 2 Spacetime decomposition;
 Ricci solitons;
 marginally trapped tube;
 locally rotationally symmetric spacetimes;
 General Relativity and Quantum Cosmology
 EPrint:
 27 pages, no figure, few typos corrected, results unchanged, Accepted for publication in IJGMMP