• 대한수학회보
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• 제45권4호
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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.

• 대한수학회논문집
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• 제27권2호
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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.

• Korean Journal of Mathematics
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• 제32권2호
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• pp.315-328
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• 2024

The aim of the paper is to study an η-Einstein soliton on (2n + 1)-dimensional (k, µ)-contact metric manifold. At first, we establish various results related to (2n + 1)-dimensional (k, µ)-contact metric manifold that exhibit an η-Einstein soliton. Next we study some curvature conditions admitting an η-Einstein soliton on (2n+1)-dimensional (k, µ)-contact metric manifold. Furthermore, we consider specific conditions associated with an η-Einstein soliton on (2n+1)-dimensional (2n+1)-dimensional (k, µ)-contact metric manifold. Finally, we show the existance of an η-Einstein soliton on (k, µ)-contact metric manifold.

• 대한수학회보
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• 제42권4호
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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.

• 대한수학회논문집
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• 제35권4호
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• pp.1283-1297
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• 2020

The goal of this paper is the introduction of hyper generalized 𝜙-recurrent (k, 𝜇)-contact metric manifolds and of quasi generalized 𝜙-recurrent (k, 𝜇)-contact metric manifolds, and the investigation of their properties. Their existence is guaranteed by examples.

• 대한수학회지
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• 제57권3호
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• pp.707-719
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• 2020

In this article we study almost contact manifolds admitting weakly Einstein metrics. We first prove that if a (2n + 1)-dimensional Sasakian manifold admits a weakly Einstein metric, then its scalar curvature s satisfies -6 ⩽ s ⩽ 6 for n = 1 and -2n(2n + 1) ${\frac{4n^2-4n+3}{4n^2-4n-1}}$ ⩽ s ⩽ 2n(2n + 1) for n ⩾ 2. Secondly, for a (2n + 1)-dimensional weakly Einstein contact metric (κ, μ)-manifold with κ < 1, we prove that it is flat or is locally isomorphic to the Lie group SU(2), SL(2), or E(1, 1) for n = 1 and that for n ⩾ 2 there are no weakly Einstein metrics on contact metric (κ, μ)-manifolds with 0 < κ < 1. For κ < 0, we get a classification of weakly Einstein contact metric (κ, μ)-manifolds. Finally, it is proved that a weakly Einstein almost cosymplectic (κ, μ)-manifold with κ < 0 is locally isomorphic to a solvable non-nilpotent Lie group.

• Kyungpook Mathematical Journal
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• 제45권2호
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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 ${\mu}=0$.

• 대한수학회논문집
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• 제35권2호
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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.

• 대한수학회논문집
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• 제31권1호
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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.