• Title, Summary, Keyword: de Sitter space

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Classical Mechanics in the De Sitter Space

  • Chung, Won Sang;Hassanabadi, Hassan
    • Journal of the Korean Physical Society
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    • v.71 no.1
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    • pp.13-18
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    • 2017
  • In this paper we consider the deformed mechanics in de Sitter space. First we discuss one dimensional mechanics in de Sitter space. We also discuss the mechanics in D-dimensional de Sitter space. As an example we discuss the deformed Kepler orbit.

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.

BJÖRLING FORMULA FOR MEAN CURVATURE ONE SURFACES IN HYPERBOLIC THREE-SPACE AND IN DE SITTER THREE-SPACE

  • Yang, Seong-Deog
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.1
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    • pp.159-175
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    • 2017
  • We solve the $Bj{\ddot{o}}rling$ problem for constant mean curvature one surfaces in hyperbolic three-space and in de Sitter three-space. That is, we show that for any regular, analytic (and spacelike in the case of de Sitter three-space) curve ${\gamma}$ and an analytic (timelike in the case of de Sitter three-space) unit vector field N along and orthogonal to ${\gamma}$, there exists a unique (spacelike in the case of de Sitter three-space) surface of constant mean curvature 1 which contains ${\gamma}$ and the unit normal of which on ${\gamma}$ is N. Some of the consequences are the planar reflection principles, and a classification of rotationally invariant CMC 1 surfaces.

SPACE-LIKE SUBMANIFOLDS WITH CONSTANT SCALAR CURVATURE IN THE DE SITTER SPACES

  • Liu, Ximin
    • Journal of the Korean Mathematical Society
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    • v.38 no.1
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    • pp.135-146
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    • 2001
  • Let M(sup)n be a space-ike submanifold in a de Sitter space M(sub)p(sup)n+p (c) with constant scalar curvature. We firstly extend Cheng-Yau's Technique to higher codimensional cases. Then we study the rigidity problem for M(sup)n with parallel normalized mean curvature vector field.

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MAXIMAL SPACE-LIKE HYPERSURFACES IN H14(-1) WITH ZERO GAUSS-KRONECKER CURVATURE

  • CHENG QING-MING;SUH YOUNG JIN
    • Journal of the Korean Mathematical Society
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    • v.43 no.1
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    • pp.147-157
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    • 2006
  • In this paper, we study complete maximal space-like hypersurfaces with constant Gauss-Kronecker curvature in an antide Sitter space $H_1^4(-1)$. It is proved that complete maximal spacelike hypersurfaces with constant Gauss-Kronecker curvature in an anti-de Sitter space $H_1^4(-1)$ are isometric to the hyperbolic cylinder $H^2(c1){\times}H^1(c2)$ with S = 3 or they satisfy $S{\leq}2$, where S denotes the squared norm of the second fundamental form.

RIGIDITY THEOREMS IN THE HYPERBOLIC SPACE

  • De Lima, Henrique Fernandes
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.1
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    • pp.97-103
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    • 2013
  • As a suitable application of the well known generalized maximum 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 $\mathbb{H}^{n+1}$. In our approach, we explore the existence of a natural duality between $\mathbb{H}^{n+1}$ and the half $\mathcal{H}^{n+1}$ of the de Sitter space $\mathbb{S}_1^{n+1}$, which models the so-called steady state space.