Curvature inheritance symmetry on M-projectively flat spacetimes

TL;DR

This paper explores the properties of curvature inheritance symmetry in M-projectively flat spacetimes, revealing its relation to conformal motion and analyzing specific cases involving perfect fluids and electromagnetic fields.

Contribution

It establishes that curvature inheritance symmetry in M-projectively flat spacetimes corresponds to conformal motion and derives conditions for perfect fluids and electromagnetic fields within this framework.

Findings

Curvature inheritance symmetry is equivalent to conformal motion in M-projectively flat spacetimes.

Such spacetimes with perfect fluids are either vacuum or have vacuum-like equations of state.

Electromagnetic field distributions do not admit curvature symmetry in this context.

Abstract

The paper aims to investigate curvature inheritance symmetry in M-projectively flat spacetimes. It is shown that the curvature inheritance symmetry in M-projectively flat spacetime is a conformal motion. We have proved that M- projective curvature tensor follows the symmetry inheritance property along a vector field $\xi$, when spacetime admits the conditions of both curvature inheritance symmetry and conformal motion or motion along the vector field $\xi$. Also, we have derived some results for M-projectively flat spacetime with perfect fluid following the Einstein field equations with a cosmological term and admitting the curvature inheritance symmetry along the vector field $\xi$. We have shown that an M-projectively flat perfect fluid spacetime obeying the Einstein field equations with a cosmological term and admitting the curvature inheritance symmetry along a vector field $\xi$ is either a vacuum or satisfies the vacuum-like equation of state. We have also shown that such spacetimes with the energy momentum tensor of an electromagnetic field distribution do not admit any curvature symmetry of general relativity. Finally, an example of M-projectively flat spacetime has been exhibited.