• Title, Summary, Keyword: Lorentzian manifold

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η-Ricci Solitons in δ-Lorentzian Trans Sasakian Manifolds with a Semi-symmetric Metric Connection

  • Siddiqi, Mohd Danish
    • Kyungpook Mathematical Journal
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    • v.59 no.3
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    • pp.537-562
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    • 2019
  • The aim of the present paper is to study the ${\delta}$-Lorentzian trans-Sasakian manifold endowed with semi-symmetric metric connections admitting ${\eta}$-Ricci Solitons and Ricci Solitons. We find expressions for the curvature tensor, the Ricci curvature tensor and the scalar curvature tensor of ${\delta}$-Lorentzian trans-Sasakian manifolds with a semisymmetric-metric connection. Also, we discuses some results on quasi-projectively flat and ${\phi}$-projectively flat manifolds endowed with a semi-symmetric-metric connection. It is shown that the manifold satisfying ${\bar{R}}.{\bar{S}}=0$, ${\bar{P}}.{\bar{S}}=0$ is an ${\eta}$-Einstein manifold. Moreover, we obtain the conditions for the ${\delta}$-Lorentzian trans-Sasakian manifolds with a semisymmetric-metric connection to be conformally flat and ${\xi}$-conformally flat.

SCREEN ISOTROPIC LEAVES ON LIGHTLIKE HYPERSURFACES OF A LORENTZIAN MANIFOLD

  • Gulbahar, Mehmet
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.429-442
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    • 2017
  • In the present paper, screen isotropic leaves on lightlike hypersurfaces of a Lorentzian manifold are introduced and studied which are inspired by the definition of isotropic immersions in the Riemannian context. Some examples of such leaves are mentioned. Furthermore, some relations involving curvature invariants are obtained.

CR-SUBMANIFOLDS OF A LORENTZIAN PARA-SASAKIAN MANIFOLD ENDOWED WITH A QUARTER SYMMETRIC METRIC CONNECTION

  • Ahmad, Mobin
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.1
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    • pp.25-32
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    • 2012
  • We define a quarter symmetric metric connection in a Lorentzia para-Sasakian manifold and study CR-submanifolds of a Lorentzian para-Sasakian manifold endowed with a quarter symmetric metric connection. Moreover, we also obtain integrability conditions of the distributions on CR-submanifolds.

ON LORENTZIAN QUASI-EINSTEIN MANIFOLDS

  • Shaikh, Absos Ali;Kim, Young-Ho;Hui, Shyamal Kumar
    • Journal of the Korean Mathematical Society
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    • v.48 no.4
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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.

SOME METRIC ON EINSTEIN LORENTZIAN WARPED PRODUCT MANIFOLDS

  • Lee, Soo-Young
    • Korean Journal of Mathematics
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    • v.27 no.4
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    • pp.1133-1147
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    • 2019
  • In this paper, let M = B×f2 F be an Einstein Lorentzian warped product manifold with 2-dimensional base. We study the geodesic completeness of some metric with constant curvature. First of all, we discuss the existence of nonconstant warping functions on M. As the results, we have some metric g admits nonconstant warping functions f. Finally, we consider the geodesic completeness on M.

NONCONSTANT WARPING FUNCTIONS ON EINSTEIN LORENTZIAN WARPED PRODUCT MANIFOLDS

  • Jung, Yoon-Tae;Choi, Eun-Hee;Lee, Soo-Young
    • Honam Mathematical Journal
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    • v.40 no.3
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    • pp.447-456
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    • 2018
  • In this paper, we consider nonconstant warping functions on Einstein Lorentzian warped product manifolds $M=B{\times}_{f^2}F$ with an 1-dimensional base B which has a negative definite metric. As the results, we discuss that on M the resulting Einstein Lorentzian warped product metric is a future (or past) geodesically complete one outside a compact set.

REMARKS ON THE TOPOLOGY OF LORENTZIAN MANIFOLDS

  • Choi, Young-Suk;Suh, Young-Jin
    • Communications of the Korean Mathematical Society
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    • v.15 no.4
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    • pp.641-648
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    • 2000
  • The purpose of this paper is to give a necessary and sufficient condition for a smooth manifold to admit a Lorentzian metric. As an application of this result, on Lorentzian manifolds we have shown that the existence of a 1-dimensional distribution is equivalent to the existence of a non-vanishing vector field.

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