Trigonometry in extended hyperbolic space and extended de Sitter space

조윤희 대한수학회 2009 대한수학회보 Vol.46 No.6

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. 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.

RIGONOMETRY IN EXTENDED HYPERBOLIC SPACE AND EXTENDED DE SITTER SPACE

Cho, Yun-Hi Korean Mathematical Society 2009 대한수학회보 Vol.46 No.6

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 Korean Mathematical Society 2013 대한수학회지 Vol.50 No.6

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.

Geometric and analytic interpretation of orthoscheme and Lambert cube in extended hyperbolic space

조윤희,김혁 대한수학회 2013 대한수학회지 Vol.50 No.6

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.

ON THE SUPREMUM OF THE TOTALLY REAL BISECTIONAL CURVATURE OF SPACE-LIKE COMPLEX SUBMANIFOLDS IN CH□(c)

Kim, Hyang Sook 인제대학교 1999 仁濟論叢 Vol.15 No.1

상수인 단면곡률(sectional curvature) c를 계량(metric)으로 가지고 개수(index)가 s인 n차원 부정치(indefinite) 켈라다양체(Kaehler manifold), 즉 n차원 부정치(indefinite)복소공간형(complex space form)Msn(C)는 c>0일 때 Psn (c), c<0일 때 Hsn (c), c=0일 때 Csn이 된다. 본 논문은 부정치 복소쌍곡공간형 Hsn (c)의 부정치 복소 부분다양체(comlpex submanifold)의 제 2기본형식(second fundamental form)의 제곱의 길이의 라프라시안(Laplacian)을 계산하고, 또 부정치 복소공간형도 Hsn (c)의 완비인 부정치 복소 부분다양체의 전실쌍단면곡률(totally real bisectional curvature)의 상한(supremum)을 계산하였다. We denote by M an n-dimensional complete space-like complex submanifold in an indefinite complex hyperbolic space CHnn+p(c). Then the supremum b(M) of the totally real bisectional curvatures of M satisfies b(M)< *****. Keywords : Indefinite complex space form, Indefinite submaifold, Space-like, Complex hyperbolic space, Totally real bisectional curvature.

STABLE MINIMAL HYPERSURFACES IN THE HYPERBOLIC SPACE

Seo, Keom-Kyo Korean Mathematical Society 2011 대한수학회지 Vol.48 No.2

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.

STABLE MINIMAL HYPERSURFACES IN THE HYPERBOLIC SPACE

서검교 대한수학회 2011 대한수학회지 Vol.48 No.2

In this paper we give an upper bound of the rst eigenvalue of the Laplace operator on a complete stable minimal hypersurface M in the hyperbolic space which has nite 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.

SOME RESULTS OF THE NEW ITERATIVE SCHEME IN HYPERBOLIC SPACE

Basarir, Metin,Sahin, Aynur Korean Mathematical Society 2017 대한수학회논문집 Vol.32 No.4

In this paper, we consider the new faster iterative scheme due to Sintunavarat and Pitea ([32]) for further investigation and we prove its strong and ${\Delta}$-convergence theorems, data dependence and stability results in hyperbolic space. Our results extend, improve and generalize several recent results in CAT(0) space and uniformly convex Banach space.

Rigidity theorems in the hyperbolic space

Henrique Fernandes de Lima 대한수학회 2013 대한수학회보 Vol.50 No.1

As a suitable application of the well known generalized max- imum principle of Omori-Yau, we obtain rigidity results concerning to a complete hypersurface immersed with bounded mean curvature in the (n+ 1)-dimensional hyperbolic space Hn+1. In our approach, we explore the existence of a natural duality between Hn+1 and the half Hn+1 of the de Sitter space Sn+1 1 , which models the so-called steady state space.

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

Cho, Yunhi Korean Mathematical Society 2013 대한수학회논문집 Vol.28 No.4

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.