• 대한수학회지
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• 제43권6호
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• pp.1143-1158
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• 2006

We consider the projectivization of Minkowski space with the analytic continuation of the hyperbolic metric and call this an extended hyperbolic space. We can measure the volume of a domain lying across the boundary of the hyperbolic space using an analytic continuation argument. In this paper we show this method can be further generalized to find the volume of a domain with smooth boundary with suitable regularity in dimension 2 and 3. We also discuss that this volume is invariant under the group of hyperbolic isometries and that this regularity condition is sharp.

• East Asian mathematical journal
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• 제37권3호
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• pp.307-317
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• 2021

In this paper, we deal with a problem to determine the type of automorphisms of the unit disc in ℂ in terms of intrinsic geometry. We will characterize the hyperbolicity and parabolicity of automorphism by the distance function of the Poincaré metric.

• 대한수학회논문집
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• 제4권2호
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• pp.265-272
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• 1989

• East Asian mathematical journal
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• 제39권1호
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• pp.11-21
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• 2023

Non-trivial automorphisms of the unit disc in the complex plane can be classified by three classes; elliptic, parabolic and hyperbolic automorphisms. This classification is due to a representation in the projective special linear group of the real field, or in terms of fixed points on the closure of the unit disc. In this paper, we will characterize this classification by the distance function of the Poincaré metric on the interior of the unit disc.

• 호남수학학술지
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• 제23권1호
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• pp.91-111
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• 2001

In this paper we prove the following : Let M be a semi-invariant submanifold with almost contact metric structure (${\phi}$, ${\xi}$, g) of codimension 3 in a complex hyperbolic space $H_{n+1}{\mathbb{C}}$. Suppose that the third fundamental form n satisfies $dn=2{\theta}{\omega}$ for a certain scalar ${\theta}({\leq}{\frac{c}{2}})$, where ${\omega}(X,\;Y)=g(X,\;{\phi}Y)$ for any vectors X and Y on M. Then M has constant eigenvalues correponding the shape operator A in the direction of the distinguished normal and the structure vector ${\xi}$ is an eigenvector of A if and only if M is locally congruent to one of the type $A_0$, $A_1$, $A_2$ or B in $H_n{\mathbb{C}}$.

• 대한수학회보
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• 제60권2호
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• pp.507-514
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• 2023

Suppose that a line passing through a given point P intersects a given circle 𝓒 at Q and R in the Euclidean plane. It is well known that |PQ||P R| is independent of the choice of the line as long as the line meets the circle at two points. It is also known that similar properties hold in the 2-sphere and in the hyperbolic plane. New proofs for the similar properties in the 2-sphere and in the hyperbolic plane are given.

• 대한수학회지
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• 제32권3호
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• pp.471-482
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• 1995

A complex n-dimensional Kaehlerian manifold of constant holomorphic sectional curvature c is called a comples space form, which is denoted by $M_n(c)$. A complete and simply connected complex space form consists of a complex projective space $P_nC$, a complex Euclidean space $C^n$ or a complex hyperbolic space $H_nC$, according as c > 0, c = 0 or c < 0. The induced almost contact metric structure of a real hypersurface M of $M_n(c)$ is denoted by $(\phi, \zeta, \eta, g)$.

• 충청수학회지
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• 제22권4호
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• pp.771-776
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• 2009

Let (H, g) denote the upper half plane in $R^2$ with the Riemannian metric g := ($(dx)^2$ + $(dy)^2$)$/y^2$. First of all we get a necessary and sufficient condition for a diffeomorphism $\phi$ of (H, g) to be a harmonic map. And, we obtain the fact that if a diffeomorphism $\phi$ of (H, g) is a harmonic function, then the following facts are equivalent: (1) $\phi$ is a harmonic map; (2) $\phi$ is an affine transformation; (3) $\phi$ is an isometry (motion).

• 충청수학회지
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• 제25권3호
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• pp.491-499
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• 2012

The hyperbolic metric is invariant under the action of M$\ddot{o}$bius maps and unbounded. For 0 < $r$ < 1, there is an r-lattice in the Bergman metric. Using this r-lattice, we get the measure ${\mu}_r$ and the Toeplitz operator $T^{\alpha}_{\mu}_r$ and we prove that $T^{\alpha}_{\mu}_r$ is bounded and $T^{\alpha}_{\mu}_r$ is compact under some condition.

• 대한수학회논문집
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• 제30권3호
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• pp.209-219
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• 2015

In this article, we first give metric version of an iteration scheme of Agarwal et al. [1] and approximate fixed points of two finite families of nonexpansive mappings in hyperbolic spaces through this iteration scheme which is independent of but faster than Mann and Ishikawa scheme. Also we consider case of three finite families of nonexpansive mappings. But, we need an extra condition to get convergence. Our convergence theorems generalize and refine many know results in the current literature.