• Journal of the Korean Mathematical Society
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• v.53 no.2
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• pp.331-346
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• 2016

The primary aim of this paper is to generalize a theorem of Hirzebruch for the complex 2-dimensional Bott manifolds, usually called Hirzebruch surfaces, to more general Bott towers of height n. To do so, we first show that all complex vector bundles of rank 2 over a Bott manifold are classified by their total Chern classes. As a consequence, in this paper we show that two Bott manifolds $B_n({\alpha}_1,{\ldots},{\alpha}_{n-1},{\alpha}_n)$ and $B_n({\alpha}_1,{\ldots},{\alpha}_{n-1},{\alpha}_n^{\prime})$ are isomorphic to each other, as Bott towers if and only if both ${\alpha}_n{\equiv}{\alpha}_n^{\prime}$ mod 2 and ${\alpha}_n^2=({\alpha}_n^{\prime})^2$ hold in the cohomology ring of $B_{n-1}({\alpha}_1,{\ldots},{\alpha}_{n-1})$ over integer coefficients. This result will complete a circle of ideas initiated in [11] by Ishida. We also give some partial affirmative remarks toward the assertion that under certain condition our main result still holds to be true for two Bott manifolds just diffeomorphic, but not necessarily isomorphic, to each other.

• Communications of the Korean Mathematical Society
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• v.10 no.1
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• pp.163-173
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• 1995

Inspired in [2,9,10,17], pp. E. Ehrlich and S. B. Kim in [4] cultivated the Riccati equation related to the Raychaudhuri equation of General Relativity for the stable Jacobi tensor along the geodesics to extend the Hawking-Penrose conjugacy theorem to $$ f(t) = Ric(c(t)',c'(t)) + tr(\sigma(A)^2) $$ where $\sigma(A)$ is the shear tensor of A for the stable Jacobi tensor A with $A(t_0) = Id$ along the complete Riemannian or complete nonspacelike geodesics c.

• The Pure and Applied Mathematics
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• v.11 no.4
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• pp.309-321
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• 2004

For a complete open Riemannian manifold, the ideal boundary consists of points at infinity. The so-called Busemann-functions play the role of distance functions for points at infinity. We study the similarity and difference between Busemann-functions and ordinary distance functions.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1461-1466
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• 2010

aSuppose that (M, g) is a complete Riemannian manifold with Ricci curvature bounded below by -K < 0 and (N, $\bar{b}$) is a complete Riemannian manifold with sectional curvature bounded above by a constant $\mu$ > 0. Let u : $M{\times}[0,\;{\infty}]{\rightarrow}B_{\tau}(p)$ is a heat equation for harmonic map. We estimate the energy density of u.

• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.853-865
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• 2019

In this paper, we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a complete Riemannian manifold: $${\Delta}_Vu+cu^{\alpha}=0$$, where c, ${\alpha}$ are two real constants and $c{\neq}0$. By applying Bochner formula and the maximum principle, we obtain local gradient estimates for positive solutions of the above equation on complete Riemannian manifolds with Bakry-${\acute{E}}mery$ Ricci curvature bounded from below, which generalize some results of [8].

• Kyungpook Mathematical Journal
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• v.64 no.3
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• pp.435-459
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• 2024

First, we give a complete classification of pseudo Ricci-Bourguignon soliton on real hypersurfaces in the complex projective space ℂPⁿ = SU_n+1/S(U₁·U_n). Next, as an application, we give a complete classification of gradient pseudo Ricci-Bourguignon soliton on real hypersurfaces in the complex projective space ℂPⁿ.

• Communications of the Korean Mathematical Society
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• v.8 no.2
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• pp.315-328
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• 1993

• Bulletin of the Korean Mathematical Society
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• v.37 no.2
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• pp.317-336
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• 2000

In this paper, when N is a compact Riemannian manifold, we discuss the method of using warped products to construct timelike or null future (or past) complete Lorentzian metrics on $M=(-{\infty},{\;}\infty){\;}{\times}f^N$ with specific scalar curvatures.

• Kyungpook Mathematical Journal
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• v.52 no.1
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• pp.1-11
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• 2012

Our attention is turned to four-dimensional pseudo-Riemannian naturally reductive homogeneous spaces. In particular, our study leads to a complete classification of them.

• Communications of the Korean Mathematical Society
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• v.13 no.1
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• pp.95-107
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• 1998

In this paper we prove the Hopf-Rinow theorem for CAT(0) spaces and show that the ideal boundaries of complete CAT(0) manifolds of dimension 2 or 3 with some additional conditions are homeomorphic to the circle or 2-sphere by the characterization of the local shadows around the branch points.