• Bulletin of the Korean Mathematical Society
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• v.43 no.4
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• pp.679-692
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• 2006

On a compact oriented n-dimensional manifold $(M^n,\;g)$, it has been conjectured that a metric g satisfying the critical point equation (2) should be Einstein. In this paper, we prove that if a manifold $(M^4,\;g)$ is a 4-dimensional oriented compact warped product, then g can not be a solution of CPE with a non-zero solution function f.

• Korean Journal of Mathematics
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• v.18 no.2
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• pp.133-139
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• 2010

The manifold $g-ESX_n$ is a generalized n-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $g_{{\lambda}{\mu}}$ through the ES-connection which is both Einstein and semi-symmetric. In this paper, we investigate the properties of the vectors $X_{\lambda}$, $S_{\lambda}$ and $U_{\lambda}$ of $g-ESX_n$, with main emphasis on the derivation of several useful generalized identities involving it.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.641-652
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• 1998

Chung and et al. ([2].1991) introduced a new concept of a manifold, denoted by $^{\ast}g-SEX_n$, in Einstein's n-dimensional $^{\ast}g$-unified field theory. The manifold $^{\ast}g-SEX_n$ is a generalized n-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $^{\ast}g^{\lambda \nu}$ through the SE-connection which is both Einstein and semi-symmetric. In this paper, they proved a necessary and sufficient condition for the unique existence of SE-connection and sufficient condition for the unique existence of SE-connection and presented a beautiful and surveyable tensorial representation of the SE-connection in terms of the tensor $^{\ast}g^{\lambda \nu}$. Recently, Chung and et al.([3],1998) obtained a concise tensorial representation of SE-curvature tensor defined by the SE-connection of $^{\ast}g-SEX_n$ and proved deveral identities involving it. This paper is a direct continuations of [3]. In this paper we derive surveyable tensorial representations of constracted curvature tensors of $^{\ast}g-SEX_n$ and prove several generalized identities involving them. In particular, the first variation of the generalized Bianchi's identity in $^{\ast}g-SEX_n$, proved in theorem (2.10a), has a great deal of useful physical applications.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1651-1670
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• 2016

In this paper, we investigate exponentially biharmonic maps u : (M, g) ${\rightarrow}$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if $\int_{M}e^{\frac{p{\mid}r(u){\mid}^2}{2}{\mid}{\tau}(u){\mid}^pdv_g$ < ${\infty}$ ($p{\geq}2$), $\int_{M}{\mid}{\tau}(u){\mid}^2dv_g$ < ${\infty}$ and $\int_{M}{\mid}d(u){\mid}^2dv_g$ < ${\infty}$, then u is harmonic. When u is an isometric immersion, we get that if $\int_{M}e^{\frac{pm^2{\mid}H{\mid}^2}{2}}{\mid}H{\mid}^qdv_g$ < ${\infty}$ for 2 ${\leq}$ p < ${\infty}$ and 0 < q ${\leq}$ p < ${\infty}$, then u is minimal. We also obtain that any weakly convex exponentially biharmonic hypersurface in space form N(c) with $c{\leq}0$ is minimal. These results give affirmative partial answer to conjecture 3 (generalized Chen's conjecture for exponentially biharmonic submanifolds).

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.301-309
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• 1995

Manifolds with recurrent connections have been studied by many authors, such as Chung, Datta, E.M.Patterson, M.Prvanovitch, Singal, and TAkano, etc (refer to [2] and [3]). Examples of such manifolds are those of recurrent curvature, Ricci-recurrent manifolds, and birecurrent manifolds.

• Korean Journal of Mathematics
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• v.21 no.1
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• pp.91-100
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• 2013

This paper is a direct continuation of [1] and [2]. In this paper we investigate some properties of ES-curvature tensor and contracted ES-curvature tensor of g - $ESX_n$. Also, we study the field equations in the n-dimensional ES manifold g - $ESX_n$.

• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.327-340
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• 2003

We are devoted to dealing with the conformal theory of a Rizza manifold with a generalized Finsler metric $G_{ij}$ (x,y) and a new conformal invariant non-linear connection $M^{i}$ $_{j}$ (x,y) constructed from the generalized Cern's non-linear connection $N^{i}$ $_{j}$ (x,y) and almost complex structure $f^{i}$ $_{j}$ (x). First, we find a conformal invariant connection ( $M_{j}$ $^{i}$ $_{k}$ , $M^{i}$ $_{j}$ , $C_{j}$ $^{i}$ $_{k}$ ) and conformal invariant tensors. Next, the nearly Kaehlerian (G, M)-structures under conformal change in a Rizza manifold are investigate.

• Korean Journal of Mathematics
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• v.29 no.2
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• pp.445-453
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• 2021

The purpose of the paper is to study of Para-Kenmotsu metric as a 𝜂-Ricci soliton. The paper is organized as follows: • If an 𝜂-Einstein para-Kenmotsu metric represents an 𝜂-Ricci soliton with flow vector field V, then it is Einstein with constant scalar curvature r = -2n(2n + 1). • If a para-Kenmotsu metric g represents an 𝜂-Ricci soliton with the flow vector field V being an infinitesimal paracontact transformation, then V is strict and the manifold is an Einstein manifold with constant scalar curvature r = -2n(2n + 1). • If a para-Kenmotsu metric g represents an 𝜂-Ricci soliton with non-zero flow vector field V being collinear with 𝜉, then the manifold is an Einstein manifold with constant scalar curvature r = -2n(2n + 1). Finally, we cited few examples to illustrate the results obtained.

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.393-400
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• 1994

Let M(f,η,ξ,g) be a (2m ＋ 1)-dimensional Sasakian manifold with soldering form dp ∈ ΓHom(Λ/sup q/TM, TM) (dp: canonical vector-valued 1-form) where f,η,ξ and g are the (1,1)-tensor field, the structure 1-form, the structure vector field and the metric tensor of M, respectively.(omitted)