• Title, Summary, Keyword: hyperbolic space

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RIGONOMETRY IN EXTENDED HYPERBOLIC SPACE AND EXTENDED DE SITTER SPACE

  • Cho, Yun-Hi
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.6
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    • pp.1099-1133
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    • 2009
  • We study the hyperbolic cosine and sine laws in the extended hyperbolic space which contains hyperbolic space as a subset and is an analytic continuation of the hyperbolic space. And we also study the spherical cosine and sine laws in the extended de Sitter space which contains de Sitter space S$^n_1$ as a subset and is also an analytic continuation of de Sitter space. In fact, the extended hyperbolic space and extended de Sitter space are the same space only differ by -1 multiple in the metric. Hence these two extended spaces clearly show and apparently explain that why many corresponding formulas in hyperbolic and spherical space are very similar each other. From these extended trigonometry laws, we can give a coherent and geometrically simple explanation for the various relations between the lengths and angles of hyperbolic polygons, and relations on de Sitter polygons which lie on S$^2_1$, and tangent laws for various polyhedra.

GEOMETRIC AND ANALYTIC INTERPRETATION OF ORTHOSCHEME AND LAMBERT CUBE IN EXTENDED HYPERBOLIC SPACE

  • Cho, Yunhi;Kim, Hyuk
    • Journal of the Korean Mathematical Society
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    • v.50 no.6
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    • pp.1223-1256
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    • 2013
  • We give a geometric proof of the analyticity of the volume of a tetrahedron in extended hyperbolic space, when vertices of the tetrahedron move continuously from inside to outside of a hyperbolic space keeping every face of the tetrahedron intersecting the hyperbolic space. Then we find a geometric and analytic interpretation of a truncated orthoscheme and Lambert cube in the hyperbolic space from the viewpoint of a tetrahedron in the extended hyperbolic space.

STABLE MINIMAL HYPERSURFACES IN THE HYPERBOLIC SPACE

  • Seo, Keom-Kyo
    • Journal of the Korean Mathematical Society
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    • v.48 no.2
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    • pp.253-266
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    • 2011
  • In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface M in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on M. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stable. We also describe stability of catenoids and helicoids in the hyperbolic space. In particular, it is shown that there exists a family of stable higher-dimensional catenoids in the hyperbolic space.

VOLUME OF C1,α-BOUNDARY DOMAIN IN EXTENDED HYPERBOLIC SPACE

  • Cho, Yun-Hi;Kim, Hyuk
    • 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.

GEOMETRIC CLASSIFICATION OF ISOMETRIES ACTING ON HYPERBOLIC 4-SPACE

  • Kim, Youngju
    • Journal of the Korean Mathematical Society
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    • v.54 no.1
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    • pp.303-317
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    • 2017
  • An isometry of hyperbolic space can be written as a composition of the reflection in the isometric sphere and two Euclidean isometries on the boundary at infinity. The isometric sphere is also used to construct the Ford fundamental domains for the action of discrete groups of isometries. In this paper, we study the isometric spheres of isometries acting on hyperbolic 4-space. This is a new phenomenon which occurs in hyperbolic 4-space that the two isometric spheres of a parabolic isometry can intersect transversally. We provide one geometric way to classify isometries of hyperbolic 4-space using the isometric spheres.

AN ELEMENTARY PROOF OF SFORZA-SANTALÓ RELATION FOR SPHERICAL AND HYPERBOLIC POLYHEDRA

  • Cho, Yunhi
    • 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.

FOCAL SURFACES AND EVOLUTES OF CURVES IN HYPERBOLIC SPACE

  • Hayashi, Ryota;Izumiya, Shyuichi;Sato, Takami
    • Communications of the Korean Mathematical Society
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    • v.32 no.1
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    • pp.147-163
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    • 2017
  • We define de Sitter focal surfaces and hyperbolic focal surfaces of hyperbolic space curves. As an application of the theory of unfoldings of function germs, we investigate the singularities of these surfaces. For characterizing the singularities of these surfaces, we discover a new hyperbolic invariants and investigate the geometric meanings.

HYPERBOLIC SPINOR DARBOUX EQUATIONS OF SPACELIKE CURVES IN MINKOWSKI 3-SPACE

  • Balci, Yakup;Erisir, Tulay;Gungor, Mehmet Ali
    • 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.

GEOMETRY ON EXOTIC HYPERBOLIC SPACES

  • Kim, In-Kang
    • 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.

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HOMOGENEOUS REAL HYPERSURFACES IN A COMPLEX HYPERBOLIC SPACE WITH FOUR CONSTANT PRINCIPAL CURVATURES

  • Song, Hyunjung
    • Journal of the Chungcheong Mathematical Society
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    • v.21 no.1
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    • pp.29-48
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    • 2008
  • We deal with the classification problem of real hypersurfaces in a complex hyperbolic space. In order to classify real hypersurfaces in a complex hyperbolic space we characterize a real hypersurface M in $H_n(\mathbb{C})$ whose structure vector field is not principal. We also construct extrinsically homogeneous real hypersurfaces with four distinct curvatures and their structure vector fields are not principal.

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