Comprehensive quasi-Einstein spacetime with application to general relativity

TL;DR

This paper introduces the comprehensive quasi-Einstein manifold C(QE)$_{n}$, extending known quasi-Einstein types, and explores their geometric and physical properties, including applications to general relativity and viscous fluid spacetimes.

Contribution

It defines the new comprehensive quasi-Einstein manifold C(QE)$_{n}$, investigates its properties, and establishes connections with conformal flatness, curvature, and physical models in general relativity.

Findings

Conformally flat C(QE)$_{n}$ is equivalent to a manifold of comprehensive quasi-constant curvature.

The paper constructs non-trivial examples of C(QE)$_{n}$ manifolds.

Analyzes the properties of C(QE)$_{4}$ in viscous fluid spacetimes.

Abstract

The aim of this paper is to extend the notion of all known quasi-Einstein manifolds like generalized quasi-Einstein, mixed generalized quasi-Einstein manifold, pseudo generalized quasi-Einstein manifold and many more and name it comprehensive quasi Einstein manifold C(QE)$_{n}$. We investigate some geometric and physical properties of the comprehensive quasi Einstein manifolds C(QE)$_{n}$ under certain conditions. We study the conformal and conharmonic mappings between C(QE)$_{n}$ manifolds. Then we examine the C(QE)$_{n}$ with harmonic Weyl tensor. We investigate geometric and physical properties of the comprehensive quasi Einstein manifolds C(QE)$_{n}$ under certain conditions. We define the manifold of comprehensive quasi-constant curvature and proved that conformally flat C(QE)$_{n}$ is manifold of comprehensive quasi-constant curvature and vice versa. We study the general two viscous fluid spacetime C(QE)$_{4}$ and find out some important consequences about C(QE)$_{4}$. We study C(QE)$_{n}$ with vanishing space matter tensor. Finally, we prove the existence of such manifolds by constructing non-trivial example.