• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.825-837
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• 2022

We characterize a three-dimensional Riemannian manifold endowed with a type of semi-symmetric metric P-connection. At first, it is proven that if the metric of such a manifold is a gradient m-quasi-Einstein metric, then either the gradient of the potential function 𝜓 is collinear with the vector field P or, λ = -(m + 2) and the manifold is of constant sectional curvature -1, provided P𝜓 ≠ m. Next, it is shown that if the metric of the manifold under consideration is a gradient 𝜌-Einstein soliton, then the gradient of the potential function is collinear with the vector field P. Also, we prove that if the metric of a 3-dimensional manifold with semi-symmetric metric P-connection is a gradient 𝜔-Ricci soliton, then the manifold is of constant sectional curvature -1 and λ + 𝜇 = -2. Finally, we consider an example to verify our results.

• Restorative Dentistry and Endodontics
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• v.30 no.4
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• pp.327-334
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• 2005

This study investigated the effect of anticurvature filing method on preparation of the curved root canal using ProFile. Thirty six resin blocks were divided equally into three groups by instrumentation motions: anticurvature filing motion. circumferential filing motion and straight up-and-down motion. Each resin block was sectioned at 8mm level from the apex and at the greatest curvature of the canal and reassembled in metal mold by a modified Bramante technique. All groups were instrumented with the ProFile system. At each levels. image of sectioned surface were taken using CCD camera under a stereomicroscope at $\times40$ magnification and stored. Distances of transportation at the inner and outer area of curvature and the centering ratio were determined and compared by statistical analysis. along with the assessment of the increase of root canal cross-sectional area. The results were as follows; 1. In all groups. there was no statistical difference in the mean increase of root canal cross-sectional area. the centering ratio. and the mean distances of transportation at the inner area of curvature at each level. 2. At 8mm level from the apex. the mean distances of transportation at the outer area of curvature decreases in following order anticurvature filing motion. circumferential filing motion. straight up-and­down motion but. no significant difference at the greatest curvature of the canal among three groups. Effect of anticurvature filing motion using ProFile does not seem to be different from other instrumentation motions at the inner area of curvature in curved root canal.

• Bulletin of the Korean Mathematical Society
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• v.34 no.2
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• pp.173-184
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• 1997

An n-dimensional complex space form $M_n(c)$ is a Kaehlerian manifold of constant holomorphic sectional curvature c. As is well known, complete and simply connected complex space forms are a complex projective space $P_n C$, a complex Euclidean space $C_n$ or a complex hyperbolic space $H_n C$ according as c > 0, c = 0 or c < 0.

• Bulletin of the Korean Mathematical Society
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• v.32 no.1
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• pp.13-18
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• 1995

In order to investigate the differential structure of a Riemannian manifold (M, g), it seems a powerful tool to study the differential structure of its tangent bundle TM. In this point of view, K. Aso [1] studied, using the Sasaki metric $\tilde{g}$, the relation between the curvature tensor on (M, g) and that on (TM, $\tilde{g}$).

• Journal of the Korean Mathematical Society
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• v.32 no.4
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• pp.835-848
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• 1995

Let $M_n$(c) be an n-dimensional complete and simply connected Kahlerian manifold of constant holomorphic sectional curvature c, which is called a complex space form. Then according to c > 0, c = 0 or c < 0 it is a complex projective space $P_nC$, a complex Euclidean space $C^n$ or a complex hyperbolic space $H_nC$.

• The Pure and Applied Mathematics
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• v.18 no.1
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• pp.79-86
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• 2011

Suppose that (M,g) is a complete and noncompact Riemannian mani-fold with Ricci curvature bounded below by $-K{\leq}0$ and (N, $\bar{g}$) is a complete Riemannian manifold with nonpositive sectional curvature. Let u : $M{\times}[0,{\infty}){\rightarrow}N$ be the solution of a heat equation for harmonic map with a bounded image. We estimate the energy density of u.

• Kyungpook Mathematical Journal
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• v.61 no.1
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• pp.191-203
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• 2021

Let M be an almost Kenmotsu manifold of dimension 2n + 1 having non-vanishing ��-sectional curvature such that trℓ > -2n - 2. We prove that any second order parallel tensor on M is a constant multiple of the associated metric tensor and obtained some consequences of this. Vector fields keeping curvature tensor invariant are characterized on M.

• Honam Mathematical Journal
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• v.44 no.1
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• pp.1-16
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• 2022

In this paper, we define and study warped product skew semi-invariant submanifolds of a locally golden Riemannian manifold. We investigate a necessary and sufficient condition for a skew semi-invariant submanifold of a locally golden Riemannian manifold to be a locally warped product. An equality between warping function and the squared normed second fundamental form of such submanifolds is established. We also construct an example of warped product skew semi-invariant submanifolds.

• Journal of the Korean Mathematical Society
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• v.60 no.1
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• pp.143-165
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• 2023

For each connected and simply connected three-dimensional non-unimodular Lie group, we classify the left invariant Lorentzian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the Ricci operator, the scalar curvature, and the sectional curvatures as functions of left invariant Lorentzian metrics on each of these groups. Our study is a continuation and extension of the previous studies done in [3] for Riemannian metrics and in [1] for Lorentzian metrics on unimodular Lie groups.

• Communications of the Korean Mathematical Society
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• v.25 no.3
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• pp.451-456
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• 2010

In this paper, we construct explicitly K$\ddot{a}$hler metrics on the ellipsoids and calculate their sectional curvatures. Using MAPLE [3], we obtain some geodesic curves on an ellipsoid so that if some conditions are dropped in Question 2.1 [4], then Question 2.1 is not true.