Characterizations of a Lorentzian Manifold with a semi-symmetric metric connection

TL;DR

This paper explores the geometric properties of Lorentzian manifolds equipped with semi-symmetric metric connections, establishing conditions under which they model perfect fluid and Robertson-Walker spacetimes, and analyzing Ricci solitons.

Contribution

It characterizes Lorentzian manifolds with semi-symmetric metric connections, linking curvature conditions to well-known spacetime models and studying Ricci solitons within this framework.

Findings

Manifolds with vanishing curvature tensor and unit time-like torse-forming vector are perfect fluid spacetimes.

Certain conditions lead to generalized Robertson-Walker spacetimes.

Manifolds with specific Ricci and torsion properties are Einstein or perfect fluid spacetimes.

Abstract

In this article, we characterize a Lorentzian manifold $\mathcal{M}$ with a semi-symmetric metric connection. At first, we consider a semi-symmetric metric connection whose curvature tensor vanishes and establish that if the associated vector field is a unit time-like torse-forming vector field, then $\mathcal{M}$ becomes a perfect fluid spacetime. Moreover, we prove that if $\mathcal{M}$ admits a semi-symmetric metric connection whose Ricci tensor is symmetric and torsion tensor is recurrent, then $\mathcal{M}$ represents a generalized Robertson-Walker spacetime. Also, we show that if the associated vector field of a semi-symmetric metric connection whose curvature tensor vanishes is a $f-$ Ric vector field, then the manifold is Einstein and if the associated vector field is a torqued vector field, then the manifold becomes a perfect fluid spacetime. Finally, we apply this connection to investigate Ricci solitons.