• 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.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.161-173
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• 2024

The main object of the present paper is to study conformal Ricci soliton on paracontact metric (k, 𝜇)-manifolds with respect to Schouten-van Kampen connection. Further, we obtain the result when paracontact metric (k, 𝜇)-manifolds with respect to Schouten-van Kampen connection satisfying the condition $^*_C({\xi},U){\cdot}^*_S=0$. Finally we characterized concircular curvature tensor on paracontact metric (k, 𝜇)-manifolds with respect to Schouten-van Kampen connection.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1535-1547
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• 2015

A vector field on a Riemannian manifold (M, g) is called concircular if it satisfies ${\nabla}X^v={\mu}X$ for any vector X tangent to M, where ${\nabla}$ is the Levi-Civita connection and ${\mu}$ is a non-trivial function on M. A smooth vector field ${\xi}$ on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation: $$\frac{1}{2}L_{\xi}g+Ric={\lambda}g$$, where $L_{\xi}g$ is the Lie-derivative of the metric tensor g with respect to ${\xi}$, Ric is the Ricci tensor of (M, g) and ${\lambda}$ is a constant. A Ricci soliton (M, g, ${\xi}$, ${\lambda}$) on a Riemannian manifold (M, g) is said to have concircular potential field if its potential field is a concircular vector field. In the first part of this paper we determine Riemannian manifolds which admit a concircular vector field. In the second part we classify Ricci solitons with concircular potential field. In the last part we prove some important properties of Ricci solitons on submanifolds of a Riemannian manifold equipped with a concircular vector field.

• Communications of the Korean Mathematical Society
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• v.24 no.3
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• pp.415-424
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• 2009

The object of the present paper is to study the generalized quasi-Einstein manifolds satisfying some conditions. Finally the existence of such manifolds is ensured by several interesting examples.

• Kyungpook Mathematical Journal
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• v.49 no.4
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• pp.789-799
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• 2009

In this study we consider ${\varphi}$-conformally flat, ${\varphi}$-conharmonically flat, ${\varphi}$-projectively at and ${\varphi}$-concircularly flat Lorentzian ${\alpha}$-Sasakian manifolds. In all cases, we get the manifold will be an ${\eta}$-Einstein manifold.

• Honam Mathematical Journal
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• v.40 no.3
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• pp.487-508
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• 2018

In this study, we use the generalized Tanaka-Webster connection on a trans-Sasakian manifold of type (${\alpha},{\beta}$) and obtain the curvature tensors of a trans-Sasakian manifold with respect to this connection. Also, we investigate some special curvature conditions of a trans-Sasakian manifold with respect to generalized Tanaka-Webster connection and finally, give an example for trans-Sasakian manifolds.

• Journal of the Korean Mathematical Society
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• v.40 no.1
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• pp.129-167
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• 2003

In this paper we introduce new notions of Ricci-like tensor and many kind of curvature-like tensors such that concircular, projective, or conformal curvature-like tensors defined on semi-Riemannian manifolds. Moreover, we give some geometric conditions which are equivalent to the Codazzi tensor, the Weyl tensor, or the second Bianchi identity concerned with such kind of curvature-like tensors respectively and also give a generalization of Weyl's Theorem given in [18] and [19].

• Honam Mathematical Journal
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• v.45 no.2
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• pp.215-230
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• 2023

In this article, generalized Sasakian space forms are investigated on W₀ -curvature tensor. Characterizations of generalized Sasakian space forms are obtained on W₀-curvature tensor. Special curvature conditions established with the help of Riemann, Ricci, concircular, projective curvature tensors are discussed on W₀-curvature tensor. With the help of these curvature conditions, important characterizations of generalized Sasakian space forms are obtained. In addition, the concepts of W₀-pseudosymmetry and W₀ -Ricci pseudosymmetry are defined and the behavior according to these concepts for the generalized Sasakian space form is examined.

• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.795-806
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• 2007

We investigate F-traceless component of the conformal curvature tensor defined by (3.6) in $K\ddot{a}hler$ manifolds of dimension ${\geq}4$, and show that the F-traceless component is invariant under concircular change. In particular, we determine $K\ddot{a}hler$ manifolds with parallel F-traceless component and improve some theorems, provided in the previous paper([2]), which are concerned with the traceless component of the conformal curvature tensor and the spectrum of the Laplacian acting on $p(0{\leq}p{\leq}2)$-forms on the manifold by using the F-traceless component.

• Korean Journal of Mathematics
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• v.27 no.3
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• pp.793-801
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• 2019

The aim of the present paper is to study certain recurrent properties of LP-Sasakian manifolds. Here we first describe Ricci ${\eta}$-recurrent LP-Sasakian manifolds. Further we study semi-generalized recurrent and three dimensional locally generalized concircularly ${\phi}$-recurrent LP-Sasakian manifolds and got interesting results.