• Kyungpook Mathematical Journal
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• v.60 no.4
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• pp.805-819
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• 2020

The objective of this study is to investigate 𝜂-Einstein solitons on (𝜀)-Kenmotsu manifolds when the Weyl-conformal curvature tensor satisfies some geometric properties such as being flat, semi-symmetric and Einstein semi-symmetric. Here, we discuss the properties of 𝜂-Einstein solitons on 𝜑-symmetric (𝜀)-Kenmotsu manifolds.

• Journal of the Korean Mathematical Society
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• v.39 no.2
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• pp.193-203
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• 2002

Weyl structure can be viewed as generalizations of Riemannian metrics. We study Weyl structures which are critical points of the squared L$^2$ norm functional of the full curvature tensor, defined on the space of Weyl structures on a compact 4-manifold. We find some relationship between these critical Weyl structures and the critical Riemannian metrics. Then in a search for homogeneous critical structures we study left-invariant metrics on some solv-manifolds and prove that they are not critical.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.227-237
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• 2018

In this paper, we study some geometric structures of a weakly cyclic Z-symmetric manifold (briefly, $[W CZS]_n$). More precisely, we prove that a conformally flat $[W CZS]_n$ satisfying certain conditions is special conformally flat and hence the manifold can be isometrically immersed in an Euclidean manifold $E^n+1$ as a hypersurface if the manifold is simply connected. Also we show that there exists a $[W CZS]_4$ with one parameter family of its associated 1-forms.

• Communications of the Korean Mathematical Society
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• v.35 no.4
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• pp.1269-1281
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• 2020

In this article, we show that a pseudo-Hermitian magnetic curve in a normal almost contact metric 3-manifold equipped with the canonical affine connection ${\hat{\nabla}}^t$ is a slant helix with pseudo-Hermitian curvature ${\hat{\kappa}}={\mid}q{\mid}\;sin\;{\theta}$ and pseudo-Hermitian torsion ${\hat{\tau}}=q\;cos\;{\theta}$. Moreover, we prove that every pseudo-Hermitian magnetic curve in normal almost contact metric 3-manifolds except quasi-Sasakian 3-manifolds is a slant helix as a Riemannian geometric sense. On the other hand we will show that a pseudo-Hermitian magnetic curve γ in a quasi-Sasakian 3-manifold M is a slant curve with curvature κ = |(t - α) cos θ + q| sin θ and torsion τ = α + {(t - α) cos θ + q} cos θ. These curves are not helices, in general. Note that if the ambient space M is an α-Sasakian 3-manifold, then γ is a slant helix.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.1003-1016
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• 2023

In this paper, we introduce the idea of twisted product lightlike submanifolds of semi-Riemannian manifolds and provide non-trivial examples of such lightlike submanifolds. Then, we prove the non-existence of proper isotropic or totally lightlike twisted product submanifolds of a semi-Riemannian manifold. We also show that for a twisted product lightlike submanifold of a semi-Riemannian manifold, the induced connection ∇ is not a metric connection. Further, we prove that a totally umbilical SCR-lightlike submanifold of an indefinite Kaehler manifold ${\tilde{M}}$ does not admit any twisted product SCR-lightlike submanifold of the type M_⊥×_ϕM_T, where M_⊥ is a totally real submanifold and MT is a holomorphic submanifold of ${\tilde{M}}$. Consequently, we obtain a geometric inequality for the second fundamental form of twisted product SCR-lightlike submanifolds of the type M_T×_ϕM_⊥ of an indefinite Kaehler manifold ${\tilde{M}}$, in terms of the gradient of ln ϕ, where ϕ stands for the twisting function. Subsequently, the equality case of this inequality is discussed. Finally, we construct a non-trivial example of a twisted product SCR-lightlike submanifold in an indefinite Kaehler manifold.

• 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.

• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.1139-1171
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• 1996

The purpose of our research is to understand geometric and topological aspects of real projective structures on surfaces. A real projective surface is a differentiable surface with an atlas of charts to $RP^2$ such that transition functions are restrictions of projective automorphisms of $RP^2$. Since such an atlas lifts projective geometry on $RP^2$ to the surface locally and consistently, one can study the global projective geometry of surfaces.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1113-1122
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• 2016

Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Yamabe flow. In the paper we derive the evolution for the first eigenvalue of geometric operator $-{\Delta}_{\phi}+{\frac{R}{2}}$ under the Yamabe flow, where ${\Delta}_{\phi}$ is the Witten-Laplacian operator, ${\phi}{\in}C^2(M)$, and R is the scalar curvature with respect to the metric g(t). As a consequence, we construct some monotonic quantities under the Yamabe flow.

• Journal of the Korean Mathematical Society
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• v.55 no.2
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• pp.391-413
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• 2018

In this paper we introduce a new tensor named semi-projective curvature tensor which generalizes the projective curvature tensor. First we deduce some basic geometric properties of semi-projective curvature tensor. Then we study pseudo semi-projective symmetric manifolds $(PSPS)_n$ which recover some known results of Chaki [5]. We provide several interesting results. Among others we prove that in a $(PSPS)_n$ if the associated vector field ${\rho}$ is a unit parallel vector field, then either the manifold reduces to a pseudosymmetric manifold or pseudo projective symmetric manifold. Moreover we deal with semi-projectively flat perfect fluid and dust fluid spacetimes respectively. As a consequence we obtain some important theorems. Next we consider the decomposability of $(PSPS)_n$. Finally, we construct a non-trivial Lorentzian metric of $(PSPS)_4$.

• Communications of the Korean Mathematical Society
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• v.36 no.4
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• pp.783-800
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• 2021

We prove the non-existence of warped product GCR-lightlike submanifolds of the type K_⊥ × _λ K_T such that K_T is a holomorphic submanifold and K_⊥ is a totally real submanifold in an indefinite Kaehler manifold $\tilde{K}$. Further, the existence of GCR-lightlike warped product submanifolds of the type K_T × _λ K_⊥ is obtained by establishing a characterization theorem in terms of the shape operator and the warping function in an indefinite Kaehler manifold. Consequently, we find some necessary and sufficient conditions for an isometrically immersed GCR-lightlike submanifold in an indefinite Kaehler manifold to be a GCR-lightlike warped product, in terms of the canonical structures f and ω. Moreover, we also derive a geometric estimate for the second fundamental form of GCR-lightlike warped product submanifolds, in terms of the Hessian of the warping function λ.