• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.965-977
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• 2016

The aim of this paper is to study the geometry of contact CR-warped product submanifolds in a cosymplectic manifold. We search several fundamental properties of contact CR-warped product submanifolds in a cosymplectic manifold. We also give necessary and sufficient conditions for a submanifold in a cosymplectic manifold to be contact CR-(warped) product submanifold. After then we establish a general inequality between the warping function and the second fundamental for a contact CR-warped product submanifold in a cosymplectic manifold and consider contact CR-warped product submanifold in a cosymplectic manifold which satisfy the equality case of the inequality and some new results are obtained.

• Honam Mathematical Journal
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• v.22 no.1
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• pp.107-111
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• 2000

We study Einstein warped product spaces. As a result, we prove the following: if M is an Einstein warped product space with base a compact 2-dimensional surface, then M is simply a Riemannian product space.

• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1641-1650
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• 2023

It is our goal in this study to present the structure of isotropic mean Berwald Finsler warped product metrics. We bring out the rich class of warped product Finsler metrics behaviour under this condition. We show that every Finsler warped product metric of dimension n ≥ 2 is of isotropic mean Berwald curvature if and only if it is a weakly Berwald metric. Also, we prove that every locally dually flat Finsler warped product metric is weakly Berwaldian. Finally, we prove that every Finsler warped product metric is of isotropic Berwald curvature if and only if it is a Berwald metric.

• Korean Journal of Mathematics
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• v.26 no.1
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• pp.75-85
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• 2018

In this paper, we study nonconstant warping functions on an Einstein warped product manifold $M=B{\times}_{f^2}F$ with a warped product metric $g=g_B+f(t)^2g_F$. And we consider a 2-dimensional base manifold B with a metric $g_B=dt^2+(f^{\prime}(t))^2du^2$. As a result, we prove the following: if M is an Einstein warped product manifold with a 2-dimensional base, then there exist generally nonconstant warping functions f(t).

• Journal of the Korean Mathematical Society
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• v.46 no.5
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• pp.979-995
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• 2009

As warped product manifolds provide an excellent setting to model space time near black holes or bodies with large gravitational field, the study of these manifolds assumes significance in general. B. Y. Chen [4] initiated the study of CR-warped product submanifolds in a Kaehler manifold. He obtained a characterization for a CR-submanifold to be locally a CR-warped product and an estimate for the squared norm of the second fundamental form of CR-warped products in a complex space form (cf [6]). In the present paper, we have obtained a necessary and sufficient conditions in terms of the canonical structures P and F on a CR-submanifold of a nearly Kaehler manifold under which the submanifold reduces to a locally CR-warped product submanifold. Moreover, an estimate for the second fundamental form of the submanifold in a generalized complex space is obtained and thus extend the results of Chen to a more general setting.

• Korean Journal of Mathematics
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• v.24 no.4
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• pp.627-636
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• 2016

In this paper, we study Ricci solitons, gradient Ricci solitons in the warped product spaces and gradient Yamabe solitons in the Riemannian product spaces. We obtain the necessary and sufficient conditions for the Riemannian product spaces to be Ricci solitons. Moreover we classify the warped product space which admit gradient Ricci solitons under some conditions of the potential function.

• Communications of the Korean Mathematical Society
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• v.34 no.4
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• pp.1303-1313
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• 2019

Recently Hui et al. ([8,9]) studied contact CR-warped product submanifolds and also warped product pseudo-slant submanifolds of a $(LCS)_n$-manifold $\bar{M}$. The characterization for both these classes of warped product submanifolds have been studied here. It is also shown that there do not exists any proper warped product bi-slant submanifold of a $(LCS)_n$-manifold. Although the existence of a bi-slant submanifold of $(LCS)_n$-manifold is ensured by an example.

• 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.

• Honam Mathematical Journal
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• v.44 no.1
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• pp.62-77
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• 2022

This study presents an 𝛼-Sasakian structure on the product manifold M₁ × 𝛽(I), where M₁ is a Kähler manifold with an exact 1-form, and 𝛽(I) is an open curve. It then defines a new type warped product metric to study the warped product of almost Hermitian manifolds with almost contact metric manifolds, contact metric manifolds, and K-contact manifolds.

• Honam Mathematical Journal
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• v.42 no.1
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• pp.151-160
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• 2020

In this paper, we obtain some necessary and sufficient conditions such that a semi-slant submanifold in a locally product Riemannian manifold reduces to a warped product semi-slant submanifold and also give an example supported this characterization.