Journal of the
Korean Mathematical Society

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008



J. Korean Math. Soc. 2011; 48(4): 669-689

Printed July 1, 2011

Copyright © The Korean Mathematical Society.

On Lorentzian quasi-Einstein manifolds

Absos Ali Shaikh, Young Ho Kim, and Shyamal Kumar Hui

University of Burdwan, Kyungpook National University, University of Burdwan


The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. The object of the present paper is to study Lorentzian quasi-Einstein manifolds. Some basic geometric properties of such a manifold are obtained. The applications of Lorentzian quasi-Einstein manifolds to the general relativity and cosmology are investigated. Theories of gravitational collapse and models of Supernova explosions [5] are based on a relativistic fluid model for the star. In the theories of galaxy formation, relativistic fluid models have been used in order to describe the evolution of perturbations of the baryon and radiation components of the cosmic medium [32]. Theories of the structure and stability of neutron stars assume that the medium can be treated as a relativistic perfectly conducting magneto fluid. Theories of relativistic stars (which would be models for supermassive stars) are also based on relativistic fluid models. The problem of accretion onto a neutron star or a black hole is usually set in the framework of relativistic fluid models. Among others it is shown that a quasi-Einstein spacetime represents perfect fluid spacetime model in cosmology and consequently such a spacetime determines the final phase in the evolution of the universe. Finally the existence of such manifolds is ensured by several examples constructed from various well known geometric structures.

Keywords: quasi-Einstein manifold, Lorentzian manifold, Lorentzian quasi-Einstein manifold, Lorentzian quasi-constant curvature, conformally flat, scalar curvature, Codazzi tensor, perfect fluid spacetime, viscous fluid spacetime, heat flux, concircular structure s

MSC numbers: 53B30, 53B50, 53C50, 53C80, 83D05