• 대한수학회보
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• 제56권5호
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• pp.1315-1325
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• 2019

The purpose of this article is to study the Ricci solitons and Ricci almost solitons on para-Kenmotsu manifold. First, we prove that if a para-Kenmotsu metric represents a Ricci soliton with the soliton vector field V is contact, then it is Einstein and the soliton is shrinking. Next, we prove that if a ${\eta}$-Einstein para-Kenmotsu metric represents a Ricci soliton, then it is Einstein with constant scalar curvature and the soliton is shrinking. Further, we prove that if a para-Kenmotsu metric represents a gradient Ricci almost soliton, then it is ${\eta}$-Einstein. This result is also hold for Ricci almost soliton if the potential vector field V is pointwise collinear with the Reeb vector field ${\xi}$.

• 호남수학학술지
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• 제45권2호
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• pp.325-339
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• 2023

The object of the present work is to study a generalized W₃ recurrent manifold. We obtain a necessary and sufficient condition for the scalar curvature to be constant in such a manifold. Also, sufficient condition for generalized W₃ recurrent manifold to be special quasi-Einstein manifold are given. Ricci symmetric and decomposable generalized W₃ recurrent manifold are studied. Finally, the existence of such a manifold is ensured by a non-trivial example.

• 대한수학회지
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• 제48권4호
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• pp.669-689
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• 2011

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.