• Title/Summary/Keyword: The $Poincar{\acute{e}}$ metric

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A CHARACTERIZATION OF AUTOMORPHISMS OF THE UNIT DISC BY THE POINCARÉ METRIC

  • Kang-Hyurk Lee;Kyu-Bo Moon
    • East Asian mathematical journal
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    • v.39 no.1
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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.

HARMONIC TRANSFORMATIONS OF THE HYPERBOLIC PLANE

  • Park, Joon-Sik
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.4
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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).

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