• 대한수학회논문집
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• 제9권3호
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• pp.641-648
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• 1994

Let M be a minimally immersed closed hypersurface in $S^{n+1}$, II the second fundamental form and $S = \Vert II \Vert^2$. It is well known that if $0 \leq S \leq n$, then $S \equiv 0$ or $S \equiv n$ and totally geodesic hypersheres and Clifford tori are the only possible minimal hypersurfaces with $S \equiv 0$ or $S \equiv n$ ([6], [2]). From these results, Chern suggested some questions on the study of compact minimal hypersurfaces on the sphere with S =constant: what are the next possible values of S to n, and does in the ambient sphere\ulcorner By the way, S is defined extrinsically but, in fact, it is an intrinsic invariant for the minimal hypersurface, i.e., S = n(n-1) - R, where R is the scalar, curvature of M. Some partial answers have been obtained for dim M = 3: Assuming $M^3 \subset S^4$ is closed and minimal with S =constant, de Almeida and Brito [1] proved that if $R \geq 0$ (or equivalently $S \leq 6$), then S = 0, 3 or 6, Peng and Terng ([5]) proved that if M has 3 distint principal curvatures, then S = 6, and in [3] Chang showed that if there exists a point which has two distinct principal curvatures, then S = 3. Hence the problem for dim M = 3 is completely done. For higher dimensional cases, not much has been known and these problems seem to be very hard without imposing some more conditions on M.

• 대한수학회보
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• 제49권2호
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• pp.435-443
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• 2012

The Yamabe invariant is a topological invariant of a smooth closed manifold, which contains information about possible scalar curvature on it. It is well-known that a product manifold $T^m{\times}B$ where $T^m$ is the m-dimensional torus, and B is a closed spin manifold with nonzero $\^{A}$-genus has zero Yamabe invariant. We generalize this to various T-structured manifolds, for example $T^m$-bundles over such B whose transition functions take values in Sp(m, $\mathbb{Z}$) (or Sp(m - 1, $\mathbb{Z}$) ${\oplus}\;{{\pm}1}$ for odd m).

• 대한수학회보
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• 제50권2호
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• pp.431-440
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• 2013

Let ($M^3$, $g$) be a 3-dimensional closed Sasakian spin manifold. Let $S_{min}$ denote the minimum of the scalar curvature of ($M^3$, $g$). Let ${\lambda}^+_1$ > 0 be the first positive eigenvalue of the Dirac operator of ($M^3$, $g$). We proved in [13] that if ${\lambda}^+_1$ belongs to the interval ${\lambda}^+_1{\in}({\frac{1}{2}},\;{\frac{5}{2}})$, then ${\lambda}^+_1$ satisfies ${\lambda}^+_1{\geq}{\frac{S_{min}+6}{8}}$. In this paper, we remove the restriction "if ${\lambda}^+_1$ belongs to the interval ${\lambda}^+_1{\in}({\frac{1}{2}},\;{\frac{5}{2}})$" and prove $${\lambda}^+_1{\geq}\;\{\frac{S_{min}+6}{8}\;for\;-\frac{3}{2}<S_{min}{\leq}30, \\{\frac{1+\sqrt{2S_{min}}+4}{2}}\;for\;S_{min}{\geq}30$$.

• 대한수학회지
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• 제54권4호
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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.

• 대한수학회보
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• 제53권3호
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• pp.875-884
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• 2016

Let x : $^{Mn-1}{\rightarrow}{\mathbb{R}}^n$ ($n{\geq}4$) be an umbilical free hyper-surface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. We denote the Laguerre scalar curvature by R and the trace-free Laguerre tensor by ${\tilde{L}}:=L-{\frac{1}{n-1}}tr(L)g$. In this paper, we prove a local classification result under the assumption of parallel Laguerre form and an inequality of the type $${\parallel}{\tilde{L}}{\parallel}{\leq}cR$$ where $c={\frac{1}{(n-3){\sqrt{(n-2)(n-1)}}}$ is appropriate real constant, depending on the dimension.

• 대한수학회보
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• 제51권2호
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• pp.531-538
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• 2014

Concentric hyperspheres in the n-dimensional Euclidean space $\mathbb{R}^n$ are the level hypersurfaces of a radial function f : $\mathbb{R}^n{\rightarrow}\mathbb{R}$. The magnitude $||{\nabla}f||$ of the gradient of such a radial function f : $\mathbb{R}^n{\rightarrow}\mathbb{R}$ is a function of the function f. We are interested in the converse problem. As a result, we show that if the magnitude of the gradient of a function f : $\mathbb{R}^n{\rightarrow}\mathbb{R}$ with isolated critical points is a function of f itself, then f is either a radial function or a function of a linear function. That is, the level hypersurfaces are either concentric hyperspheres or parallel hyperplanes. As a corollary, we see that if the magnitude of a conservative vector field with isolated singularities on $\mathbb{R}^n$ is a function of its scalar potential, then either it is a central vector field or it has constant direction.

• 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.

• 호남수학학술지
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• 제42권3호
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• pp.521-536
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• 2020

The object of the present paper is to study of Almost Quasi-Yamabe solitons and gradient almost quasi-Yamabe solitons on an Lorentzian concircular structure manifolds briefly say (LCS)_n-manifolds under infinitesimal CL-transformations and obtained sufficient conditions for such solitons to be expanding, steady and shrinking. Also we obtained a necessary and sufficient condition of an almost quasi-Yamabe soliton with respect to the CL-connection to be an almost quasi-Yamabe soliton on (LCS)_n-manifolds with respect to Levi-Civita connection. Finally, we construct an example of steady almost quasi-Yamabe soliton on 3-dimensional (LCS)_n-manifolds.

• Kyungpook Mathematical Journal
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• 제60권3호
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• pp.571-584
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• 2020

The object of the present paper is to characterize generalized Ricci recurrent (GR₄) spacetimes. Among others things, it is proved that a conformally flat GR₄ spacetime is a perfect fluid spacetime. We also prove that a GR₄ spacetime with a Codazzi type Ricci tensor is a generalized Robertson Walker spacetime with Einstein fiber. We further show that in a GR₄ spacetime with constant scalar curvature the energy momentum tensor is semisymmetric. Further, we obtain several corollaries. Finally, we cite some examples which are sufficient to demonstrate that the GR₄ spacetime is non-empty and a GR₄ spacetime is not a trivial case.

• 대한수학회지
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• 제44권4호
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• pp.941-947
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• 2007

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of nonpositive scalar curvature. Let ${\mu}0$ be the least eigenvalue of the Laplacian acting on $L^2-functions$ on M. We show that if $Ric^M{\ge}-{\mu}0$ at all $x{\in}M$ and either $Ric^M>-{\mu}0$ at some point x0 or Vol(M) is infinite, then every harmonic morphism ${\phi}:M{\to}N$ of finite energy is constant.