• Communications of the Korean Mathematical Society
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• v.28 no.4
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• pp.799-807
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• 2013

We defined and studied a naturally extended hyperbolic space (see [1] and [2]). In this study, we describe Sforza's formula [7] and Santal$\acute{o}$'s formula [6], which were rediscovered and later discussed by many mathematicians (Milnor [4], Su$\acute{a}$rez-Peir$\acute{o}$ [8], J. Murakami and Ushijima [5], and Mednykh [3]) in the spherical space in an elementary way. Thereafter, using the extended hyperbolic space, we apply the same method to prove their results in the hyperbolic space.

• Journal of the Korean Mathematical Society
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• v.36 no.3
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• pp.621-631
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• 1999

In this paper we briefly describe the geometry of the Cayley hyperbolic plane and we show that every uniform lattice in quaternionic space cannot be deformed in the Cayley hyperbolic 2-plane. We also describe the nongeometric bending deformation by developing the theory of the Cartan angular invariant for quaternionic hyperbolic space.

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.721-727
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• 2021

The convex hull of a generic triple of boundary points has non-zero finite volume in complex hyperbolic 2-space.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.4
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• pp.525-535
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• 2015

In this paper, we study on spinors with two hyperbolic components. Firstly, we express the hyperbolic spinor representation of a spacelike curve dened on an oriented (spacelike or time-like) surface in Minkowski space ${\mathbb{R}}^3_1$. Then, we obtain the relation between the hyperbolic spinor representation of the Frenet frame of the spacelike curve on oriented surface and Darboux frame of the surface on the same points. Finally, we give one example about these hyperbolic spinors.

• Journal of the Korean Mathematical Society
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• v.43 no.6
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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.

• Journal of the Korean Mathematical Society
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• v.40 no.4
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• pp.595-607
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• 2003

In the previous paper [6], the author constructed hyperbolic hypersurfaces of degree $2^{n}$ in the n-dimensional complex projective space for every $n\;\geq\;3$. The purpose of this paper is to give some improvement of this result and to show some general methods of constructions of hyperbolic hypersurfaces of higher degree, which enable us to construct hyperbolic hypersurfaces of degree d in the n-dimensional complex projective space for every $d\;\geq\;2\;{\times}\;6^{n}$.

• Journal of the Korean Mathematical Society
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• v.44 no.5
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• pp.1093-1102
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• 2007

In this note we survey some recent results on symmetric space, and related topics like rigidity, flexibility and geometry-topology aspect of symmetric space, specially in the line of hyperbolic 3-manifolds.

• Journal of the Korean Mathematical Society
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• v.46 no.3
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• pp.513-521
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• 2009

We first introduce a complex hyperbolic space and a complex hyperbolic manifold. After defining the canonical horoball and the canonical cusp on the complex hyperbolic manifold, we estimate the volumes of canonical cusps of complex hyperbolic manifolds. Finally, we deal with cusped, complex hyperbolic 2-manifolds, and in particular, the ones with only one cusp.

• Journal of the Korean Mathematical Society
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• v.50 no.2
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• pp.425-444
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• 2013

We construct some hyperbolic hyperelliptic space forms whose fundamental groups are generated by only two or three isometries. Each occurring group is obtained from a supergroup, which is an extended Coxeter group generated by plane re ections and half-turns. Then we describe covering properties and determine the isometry groups of the constructed manifolds. Furthermore, we give an explicit construction of space form of the second smallest volume nonorientable hyperbolic 3-manifold with one cusp.

• Bulletin of the Korean Mathematical Society
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• v.35 no.3
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• pp.503-514
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• 1998

We want to treat this problem for real hypersurfaces in a complex hyperbolic space $J_n(C)$. Thus it seems to be natural to consider some problems concerned with the estimation of the Ricci tensor for real hypersurfaces in $H_n(C)$. In this paper we will find a new tensorial formula concerned with the Ricci tensor and give it a characterization of horospheres and geodesic hyperspheres in a complex hyperbolic space $H_n(C)$.