• Korean Journal of Mathematics
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• v.28 no.4
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• pp.847-863
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• 2020

The aim of the present paper is to study some solitons on three dimensional generalized (𝜅, 𝜇)-contact metric manifolds. We study gradient Yamabe solitons on three dimensional generalized (𝜅, 𝜇)-contact metric manifolds. It is proved that if the metric of a three dimensional generalized (𝜅, 𝜇)-contact metric manifold is gradient Einstein soliton then ${\mu}={\frac{2{\kappa}}{{\kappa}-2}}$. It is shown that if the metric of a three dimensional generalized (𝜅, 𝜇)-contact metric manifold is closed m-quasi Einstein metric then ${\kappa}={\frac{\lambda}{m+2}}$ and 𝜇 = 0. We also study conformal gradient Ricci solitons on three dimensional generalized (𝜅, 𝜇)-contact metric manifolds.

• Communications of the Korean Mathematical Society
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• v.27 no.2
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• pp.327-339
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• 2012

In this paper we study $h$-projectively semisymmetric, ${\phi}$-pro-jectively semisymmetric, $h$-Weyl semisymmetric and ${\phi}$-Weyl semisym- metric non-Sasakian ($k$, ${\mu}$)-contact metric manifolds. In all the cases the manifold becomes an ${\eta}$-Einstein manifold. As a consequence of these results we obtain that if a 3-dimensional non-Sasakian ($k$, ${\mu}$)-contact metric manifold satisfies such curvature conditions, then the manifold reduces to an N($k$)-contact metric manifold.

• Bulletin of the Korean Mathematical Society
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• v.45 no.4
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• pp.689-700
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• 2008

In this paper we prove that a $\phi$-recurrent (k, $\mu$)-contact metric manifold is an $\eta$-Einstein manifold with constant coefficients. Next, we prove that a three-dimensional locally $\phi$-recurrent (k, $\mu$)-contact metric manifold is the space of constant curvature. The existence of $\phi$-recurrent (k, $\mu$)-manifold is proved by a non-trivial example.

• Journal of the Korean Mathematical Society
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• v.55 no.1
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• pp.161-174
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• 2018

The aim of this article is to study the k-almost Ricci soliton and k-almost gradient Ricci soliton on contact metric manifold. First, we prove that if a compact K-contact metric is a k-almost gradient Ricci soliton, then it is isometric to a unit sphere $S^{2n+1}$. Next, we extend this result on a compact k-almost Ricci soliton when the flow vector field X is contact. Finally, we study some special types of k-almost Ricci solitons where the potential vector field X is point wise collinear with the Reeb vector field ${\xi}$ of the contact metric structure.

• Bulletin of the Korean Mathematical Society
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• v.42 no.4
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• pp.713-724
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• 2005

We prove that a (k, $\mu$)-manifold with vanishing E­Bochner curvature tensor is a Sasakian manifold. Several interesting corollaries of this result are drawn. Non-Sasakian (k, $\mu$)­manifolds with C-Bochner curvature tensor B satisfying B $(\xi,\;X)\;\cdot$ S = 0, where S is the Ricci tensor, are classified. N(K)-contact metric manifolds $M^{2n+1}$, satisfying B $(\xi,\;X)\;\cdot$ R = 0 or B $(\xi,\;X)\;\cdot$ B = 0 are classified and studied.

• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.639-651
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• 2020

Consider a gradient Einstein-type metric in the setting of K-contact manifolds and (κ, µ)-contact manifolds. First, it is proved that, if a complete K-contact manifold admits a gradient Einstein-type metric, then M is compact, Einstein, Sasakian and isometric to the unit sphere 𝕊²ⁿ⁺¹. Next, it is proved that, if a non-Sasakian (κ, µ)-contact manifolds admits a gradient Einstein-type metric, then it is flat in dimension 3, and for higher dimension, M is locally isometric to the product of a Euclidean space 𝔼ⁿ⁺¹ and a sphere 𝕊ⁿ(4) of constant curvature +4.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.231-240
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• 2005

The paper shows that a hypersurface with constant curvature with the condition $hD{\subset}D$, of a contact metric manifold with a nullity condition and ${\varphi}-constant$ sectional curvature, has the curvature equals to k and ${\mu}=0$.

• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.741-751
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• 2022

The object of the present paper is to study some geometric properties of invariant submanifolds of N(k)-contact metric manifold admitting generalized Tanaka-Webster connection.

• 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.

• Communications of the Korean Mathematical Society
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• v.13 no.2
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• pp.337-343
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• 1998

In this paper we show that every contact metric manifold with vanishing contact conformal curvature tensor is a Sasakian space form.