• The Pure and Applied Mathematics
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• v.28 no.4
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• pp.297-314
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• 2021

This manuscript is divided into three segments. In the first segment, we formulate a unique common fixed point theorem satisfying generalized Meir-Keeler contraction on partially ordered metric spaces and also give an example to demonstrate the usability of our result. In the second segment of the article, some common coupled fixed point results are derived from our main results. In the last segment, we investigate the solution of some periodic boundary value problems. Our results generalize, extend and improve several well-known results of the existing literature.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.763-774
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• 2023

If a paraSasakian manifold of dimension (2n + 1) represents Bach almost solitons, then the Bach tensor is a scalar multiple of the metric tensor and the manifold is of constant scalar curvature. Additionally it is shown that the Ricci operator of the metric g has a constant norm. Next, we characterize 3-dimensional paraSasakian manifolds admitting Bach almost solitons and it is proven that if a 3-dimensional paraSasakian manifold admits Bach almost solitons, then the manifold is of constant scalar curvature. Moreover, in dimension 3 the Bach almost solitons are steady if r = -6; shrinking if r > -6; expanding if r < -6.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.195-205
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• 2024

In this paper, we consider the set of periodic shadowable points for homeomorphisms of a compact metric space, and we prove that this set satisfies some properties such as invariance and being a G_δ set. Then we investigate implication relations related to sets consisting of shadowable points, periodic shadowable points and uniformly expansive points, respectively. Assume that the set of periodic points and the set of periodic shadowable points of a homeomorphism on a compact metric space are dense in X. Then we show that a homeomorphism has the periodic shadowing property if and only if so is the restricted map to the set of periodic shadowable points. We also give some examples related to our results.

• The Pure and Applied Mathematics
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• v.31 no.1
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• pp.83-102
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• 2024

In this paper, first we establish a unique common fixed point theorem satisfying new contractive condition on partially ordered non-Archimedean fuzzy metric spaces and give an example to support our result. By using the result established in the first section of the manuscript, we formulate a unique common coupled fixed point theorem and also give an example to validate our result. In the end, we study the existence of solution of integral equation to verify our hypothesis. These results generalize, improve and fuzzify several well-known results in the existing literature.

• The Pure and Applied Mathematics
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• v.26 no.3
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• pp.111-131
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• 2019

We introduce (CLRg) property for hybrid pair $F:X{\times}X{\rightarrow}2^X$ and $g:X{\rightarrow}X$. We also introduce joint common limit range (JCLR) property for two hybrid pairs $F,G:X{\times}X{\rightarrow}2^X$ and $f,g:X{\rightarrow}X$. We also establish some common coupled fixed point theorems for hybrid pair of mappings under generalized (${\psi},{\theta},{\varphi}$)-contraction on a noncomplete metric space, which is not partially ordered. It is to be noted that to find coupled coincidence point, we do not employ the condition of continuity of any mapping involved therein. As an application, we study the existence and uniqueness of the solution to an integral equation. We also give an example to demonstrate the degree of validity of our hypothesis. The results we obtain generalize, extend and improve several recent results in the existing literature.

• Bulletin of the Korean Mathematical Society
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• v.29 no.2
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• pp.237-243
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• 1992

Let (M, g, J) be a closed Kahler manifold of complex dimension m > 1. We denote by Spec(M,g) the spectrum of the real Laplace-Beltrami operator. DELTA. acting on functions on M. The following characterization problem on the spectral rigidity of the complex projective space (CP$^{m}$ , g$_{0}$ , J$_{0}$ ) with the standard complex structure J$_{0}$ and the Fubini-Study metric g$_{0}$ has been attacked by many mathematicians : if (M,g,J) and (CP$^{m}$ ,g$_{0}$ ,J$_{0}$ ) are isospectral then is it true that (M,g,J) is holomorphically isometric to (CP$^{m}$ ,g$_{0}$ ,J$_{0}$ )\ulcorner In [BGM], [LB], it is proved that if (M,J) is (CP$^{m}$ , J$_{0}$ ) then the answer to the problem is affirmative. Tanno ([Ta]) has proved that the answer is affirmative if m .leq. 6. Recently, Wu([Wu]) has showed in a more general sense that if (M, g) and (CP$^{m}$ ,g$_{0}$ ) are (-4/m)-isospectral, m .geq. 4, and if the second betti number b$_{2}$(M) is equal to b$_{2}$(CP$^{m}$ ).

• 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.

• Journal of the Korean Mathematical Society
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• v.54 no.4
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• pp.1189-1208
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• 2017

In this paper, we study vanishing properties for $L^2$ harmonic 1-forms on a gradient shrinking Ricci soliton. We prove that if (M, g, f) is a complete oriented noncompact gradient shrinking Ricci soliton with potential function f, then there are no non-trivial $L^2$ harmonic 1-forms which are orthogonal to df. Second, we show that if the scalar curvature of the metric g is greater than or equal to (n - 2)/2, then there are no non-trivial $L^2$ harmonic 1-forms on (M, g). We also show that any multiplication of the total differential df by a function cannot be an $L^2$ harmonic 1-form unless it is trivial. Finally, we derive various integral properties involving the potential function f and $L^2$ harmonic 1-forms, and handle their applications.

• Journal of the Korean Mathematical Society
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• v.32 no.1
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• pp.51-60
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• 1995

Let $(M,g_M,F)$ be a (p+q)-dimensional connected Riemannian manifold with a foliation $F$ of codimension q and a complete bundle-like metric $g_M$ with respect to $F$. Let $Ric_D$ be the transverse Ricci field of $F$ with respect to the transverse Riemannian connection D which is a torsion-free and $g_Q$-metrical connection on the normal bundle Q of $F$. We consider transverse confomal (or, projective) fields of $F$. It is clear that a tranverse Killing field s of $F$ preserves the transverse Ricci field of $F$, that is, $\Theta(s)Ric_D = 0$, where $\Theta(s)$ denotes the transverse Lie differentiation with respect to s.

• Bulletin of the Korean Mathematical Society
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• v.41 no.1
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• pp.117-123
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• 2004

It has been conjectured that, on a compact orient able manifold M, a critical point of the total scalar curvature functional restricted the space of unit volume metrics of constant scalar curvature is Einstein. In this paper we show that if a manifold is a 3-dimensional warped product, then (M, g) cannot be a critical point unless it is isometric to the standard sphere.