• Honam Mathematical Journal
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• v.36 no.4
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• pp.795-803
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• 2014

In this paper, we decompose the curvature tensor (field) on the Abbena-Thurston manifold into three curvature-like tensor fields, and investigate geometric properties.

• Communications of the Korean Mathematical Society
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• v.34 no.4
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• pp.1315-1328
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• 2019

The object of the present paper is to study ${\eta}$-recurrent ${\ast}$-Ricci tensor, ${\ast}$-Ricci semisymmetric and globally ${\varphi}-{\ast}$-Ricci symmetric contact metric generalized (${\kappa}$, ${\mu}$)-space form. Besides these, ${\ast}$-Ricci soliton and gradient ${\ast}$-Ricci soliton in contact metric generalized (${\kappa}$, ${\mu}$)-space form have been studied.

• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.1089-1103
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• 2010

In this paper, we study lightlike submanifolds of a semi-Riemannian manifold admitting a semi-symmetric non-metric connection. We obtain a necessary and a sufficient condition for integrability of the screen distribution. Then we give the conditions under which the Ricci tensor of a lightlike submanifold with a semi-symmetric non-metric connection is symmetric. Finally, we show that the Ricci tensor of a lightlike submanifold of semi-Riemannian space form is not parallel with respect to the semi-symmetric non-metric connection.

• Honam Mathematical Journal
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• v.27 no.3
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• pp.487-504
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• 2005

Let M be a real hypersurface with almost contact metric structure (${\phi}$, ${\xi}$, ${\eta}$, g) in a nonflat complex space form $M_n(c)$. We denote by A and S be the shape operator and the Ricci tensor of M respectively. In the present paper we investigate real hypersurfaces with $g(SA{\xi},\;A{\xi})=const$. of $M_n(c)$ whose structure Jacobi operator $R_{\xi}$ commute with both ${\phi}$ and S. We give a characterization of Hopf hypersurfaces of $M_n(c)$.

• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.603-611
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• 2020

The purpose of this note is to introduce a type of Riemannian manifold called an almost quasi Ricci symmetric manifold and investigate the several properties of such a manifold on which some geometric conditions are imposed. And the existence of such a manifold is ensured by a proper example.

• Kyungpook Mathematical Journal
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• v.60 no.3
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• pp.571-584
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• 2020

The object of the present paper is to characterize generalized Ricci recurrent (GR₄) spacetimes. Among others things, it is proved that a conformally flat GR₄ spacetime is a perfect fluid spacetime. We also prove that a GR₄ spacetime with a Codazzi type Ricci tensor is a generalized Robertson Walker spacetime with Einstein fiber. We further show that in a GR₄ spacetime with constant scalar curvature the energy momentum tensor is semisymmetric. Further, we obtain several corollaries. Finally, we cite some examples which are sufficient to demonstrate that the GR₄ spacetime is non-empty and a GR₄ spacetime is not a trivial case.

• Communications of the Korean Mathematical Society
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• v.31 no.1
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• pp.163-176
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• 2016

The object of the present paper is to introduce a new curvature tensor, named generalized quasi-conformal curvature tensor which bridges conformal curvature tensor, concircular curvature tensor, projective curvature tensor and conharmonic curvature tensor. Flatness and symmetric properties of generalized quasi-conformal curvature tensor are studied in the frame of (k, ${\mu}$)-contact metric manifolds.

• Bulletin of the Korean Mathematical Society
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• v.39 no.1
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• pp.133-140
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• 2002

On compact complex hyperbolic manifolds of complex dimension two, we show that the dimension of the space of infinitesimal deformations of self-dual conformal structures is smaller than that of the deformation obstruction space and that every self-dual metric with covariantly constant Ricci tensor must be a standard one upto rescalings and diffeomorphisms.

• Communications of the Korean Mathematical Society
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• v.27 no.4
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• pp.781-793
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• 2012

We study some properties of a semi-Riemannian submanifold of a semi-Riemannian manifold with a semi-symmetric non-metric connection. Then, we prove that the Ricci tensor of a semi-Riemannian submanifold of a semi-Riemannian space form admitting a semi-symmetric non-metric connection is symmetric but is not parallel. Last, we give the conditions under which a totally umbilical semi-Riemannian submanifold with a semi-symmetric non-metric connection is projectively flat.

• Honam Mathematical Journal
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• v.28 no.4
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• pp.573-589
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• 2006

Let M be a real hypersurface with almost contact metric structure $({\phi},{\xi},{\eta},g)$ in a non flat complex space form $M_n(c)$. In this paper, we prove that if the structure Jacobi operator $R_{\xi}$ is ${\xi}$-parallel and the Ricci tensor S commutes with the structure operator $\phi$, then a real hypersurface in $M_n(c)$ is a Hopf hypersurface. Further, we characterize such Hopf hypersurface in $M_n(c)$.