• Korean Journal of Mathematics
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• v.31 no.2
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• pp.139-144
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• 2023

In this paper, we show that a recurrent manifold with harmonic curvature tensor is locally symmetric and that an Einstein and conformally recurrent manifold is locally symmetric. As a consequence, Einstein and recurrent manifolds must be locally symmetric. On the other hand, we have obtained some results for a (conformally) recurrent manifold with parallel vector field and also investigated some results for a (conformally) recurrent manifold with concircular vector field.

• The Pure and Applied Mathematics
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• v.21 no.1
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• pp.1-10
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• 2014

In this paper, we study the curvature of locally symmetric or semi-symmetric half lightlike submanifolds M of an indefinite Kenmotsu manifold $\bar{M}$, whose structure vector field is tangent to M. After that, we study the existence of the totally geodesic screen distribution of half lightlike submanifolds of indefinite Kenmotsu manifolds with parallel co-screen distribution subject to the conditions: (1) M is locally symmetric, or (2) the lightlike transversal connection is flat.

• Bulletin of the Korean Mathematical Society
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• v.49 no.2
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• pp.271-284
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• 2012

Let M be a linear Weingarten spacelike hypersurface in a locally symmetric Lorentz space with R = aH + b, where R and H are the normalized scalar curvature and the mean curvature, respectively. In this paper, we give some conditions for the complete hypersurface M to be totally umbilical.

• Honam Mathematical Journal
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• v.42 no.4
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• pp.811-819
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• 2020

We give characterizations of locally symmetric spaces M via the structural operator $h={\frac{1}{2}{\mathcal{L}}_{\xi}{\phi}}$ or the characteristic Jacobi operator ℓ = R(·, ξ)ξ on the unit tangent sphere bundles T1M.

• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.789-798
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• 2018

Let M be a compact almost $coK{\ddot{a}}hler$ 5-manifold with $K{\ddot{a}}hlerian$ leaves. In this paper, we prove that M is locally symmetric if and only if it is locally isometric to a Riemannian product of a unit circle $S^1$ and a locally symmetric compact $K{\ddot{a}}hler$ 4-manifold.

• Honam Mathematical Journal
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• v.36 no.4
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• pp.707-722
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• 2014

In this paper, we study the forms of the curvatures of lightlike submanifolds M of an indefinite Kenmotsu manifold $\bar{M}$ subject to the conditions: M is locally symmetric or M is semi-symmetric.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1259-1280
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• 2016

The paper deals with lightlike hypersurfaces which are locally symmetric, semi-symmetric and Ricci semi-symmetric in indefinite Kenmotsu manifold having constant $\bar{\phi}$-holomorphic sectional curvature c. We obtain that these hypersurfaces are totally goedesic under certain conditions. The non-existence condition of locally symmetric lightlike hyper-surfaces are given. Some Theorems of specific lightlike hypersurfaces are established. We prove, under a certain condition, that in lightlike hyper-surfaces of an indefinite Kenmotsu space form, tangent to the structure vector field, the parallel, semi-parallel, local symmetry, semi-symmetry and Ricci semi-symmetry notions are equivalent.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.135-142
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• 2013

In the present paper, we discuss the rigidity phenomenon of closed minimal submanifolds in a locally symmetric Riemannian manifold with pinched sectional curvature. We show that if the sectional curvature of the submanifold is no less than an explicitly given constant, then either the submanifold is totally geodesic, or the ambient space is a sphere and the submanifold is isometric to a product of two spheres or the Veronese surface in $S^4$.

• Journal of the Korean Mathematical Society
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• v.39 no.2
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• pp.253-275
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• 2002

The purpose of this paper is to study complete complex submanifolds of a locally symmetric Kaehler manifold with non-positive totally real normal bisectional curvature and k-pinched totally real tangent bisectional curvature.

• Communications of the Korean Mathematical Society
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• v.32 no.2
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• pp.401-416
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• 2017

The aim of this paper is to investigate locally ${\phi}-conformally$ symmetric almost Kenmotsu manifolds with its characteristic vector field ${\xi}$ belonging to some nullity distributions. Also, we give an example of a 5-dimensional almost Kenmotsu manifold such that ${\xi}$ belongs to the $(k,\;{\mu})^{\prime}$-nullity distribution and $h^{\prime}{\neq}0$.