Lorentzian manifolds equipped with a concircularly semi-symmetric metric connection

TL;DR

This paper explores Lorentzian manifolds with a concircularly semi-symmetric metric connection, showing they reduce to GRW space-times under certain conditions and analyzing their curvature and symmetry properties.

Contribution

It characterizes Lorentzian manifolds with a concircularly semi-symmetric metric connection, linking them to GRW space-times and analyzing their curvature and Einstein conditions.

Findings

Manifold reduces to GRW space-time when generator is a unit timelike vector.

Derived conditions for the manifold to be Einstein.

Proved that certain space-times are Ricci pseudo-symmetric of constant type.

Abstract

Building upon previous works characterizing GRW space-times using concircular and torse-forming vectors, this paper investigates a Lorentzian manifold equipped with a concircularly semi-symmetric metric connection. We demonstrate that such a manifold reduces to a GRW space-time under specific conditions: when the generator of the observed connection is a unit timelike vector. Also, in that case, the mentioned connection becomes a semi-symmetric metric $P$-connection. The non-zero nature of the three curvature tensors and their corresponding Ricci tensors motivates an exploration of manifold symmetries. In this way, we derive necessary and sufficient conditions for the manifold to be Einstein and we prove that a perfect fluid space-time with a semi-symmetric metric $P$-connection is Ricci pseudo-symmetric manifold of constant type. Furthermore, we show that if this space-time satisfies the Einstein's field equations without the cosmological constant, the strong energy condition is violated.