• Title, Summary, Keyword: metric group

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THE TRANSFORMATION GROUPS AND THE ISOMETRY GROUPS

  • Kim, Young-Wook
    • Bulletin of the Korean Mathematical Society
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    • v.26 no.1
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    • pp.47-52
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    • 1989
  • Methods of Riemannian geometry has played an important role in the study of compact transformation groups. Every effective action of a compact Lie group on a differential manifold leaves a Riemannian metric invariant and the study of such actions reduces to the one involving the group of isometries of a Riemannian metric on the manifold which is, a priori, a Lie group under the compact open topology. Once an action of a compact Lie group is given an invariant metric is easily constructed by the averaging method and the Lie group is naturally imbedded in the group of isometries as a Lie subgroup. But usually this invariant metric has more symmetries than those given by the original action. Therefore the first question one may ask is when one can find a Riemannian metric so that the given action coincides with the action of the full group of isometries. This seems to be a difficult question to answer which depends very much on the orbit structure and the group itself. In this paper we give a sufficient condition that a subgroup action of a compact Lie group has an invariant metric which is not invariant under the full action of the group and figure out some aspects of the action and the orbit structure regarding the invariant Riemannian metric. In fact, according to our results, this is possible if there is a larger transformation group, containing the oringnal action and either having larger orbit somewhere or having exactly the same orbit structure but with an orbit on which a Riemannian metric is ivariant under the orginal action of the group and not under that of the larger one. Recently R. Saerens and W. Zame showed that every compact Lie group can be realized as the full group of isometries of Riemannian metric. [SZ] This answers a question closely related to ours but the situation turns out to be quite different in the two problems.

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HARMONIC HOMOMORPHISMS BETWEEN TWO LIE GROUPS

  • Son, Heui-Sang;Kim, Hyun Woong;Park, Joon-Sik
    • Honam Mathematical Journal
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    • v.38 no.1
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    • pp.1-8
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    • 2016
  • In this paper, we get a complete condition for a group homomorphism of a compact Lie group with an arbitrarily given left invariant Riemannian metric into another Lie group with a left invariant metric to be a harmonic map, and then obtain a necessary and sufficient condition for a group homomorphism of (SU(2), g) with a left invariant metric g into the Heisenberg group (H, $h_0$) to be a harmonic map.

DIFFERENTIAL GEOMETRIC PROPERTIES ON THE HEISENBERG GROUP

  • Park, Joon-Sik
    • Journal of the Korean Mathematical Society
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    • v.53 no.5
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    • pp.1149-1165
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    • 2016
  • In this paper, we show that there exists no left invariant Riemannian metric h on the Heisenberg group H such that (H, h) is a symmetric Riemannian manifold, and there does not exist any H-invariant metric $\bar{h}$ on the Heisenberg manifold $H/{\Gamma}$ such that the Riemannian connection on ($H/{\Gamma},\bar{h}$) is a Yang-Mills connection. Moreover, we get necessary and sufficient conditions for a group homomorphism of (SU(2), g) with an arbitrarily given left invariant metric g into (H, h) with an arbitrarily given left invariant metric h to be a harmonic and an affine map, and get the totality of harmonic maps of (SU(2), g) into H with a left invariant metric, and then show the fact that any affine map of (SU(2), g) into H, equipped with a properly given left invariant metric on H, does not exist.

YANG-MILLS CONNECTIONS ON A COMPACT CONNECTED SEMISIMPLE LIE GROUP

  • Park, Joon-Sik
    • East Asian mathematical journal
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    • v.26 no.1
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    • pp.75-79
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    • 2010
  • Let G be a compact connected semisimple Lie group, g the Lie algebra of G, g the canonical metric (the biinvariant Riemannian metric which is induced from the Killing form of g), and $\nabla$ be the Levi-Civita connection for the metric g. Then, we get the fact that the Levi-Civita connection $\nabla$ in the tangent bundle TG over (G, g) is a Yang-Mills connection.

ON WEAKLY EINSTEIN ALMOST CONTACT MANIFOLDS

  • Chen, Xiaomin
    • Journal of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.707-719
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    • 2020
  • In this article we study almost contact manifolds admitting weakly Einstein metrics. We first prove that if a (2n + 1)-dimensional Sasakian manifold admits a weakly Einstein metric, then its scalar curvature s satisfies -6 ⩽ s ⩽ 6 for n = 1 and -2n(2n + 1) ${\frac{4n^2-4n+3}{4n^2-4n-1}}$ ⩽ s ⩽ 2n(2n + 1) for n ⩾ 2. Secondly, for a (2n + 1)-dimensional weakly Einstein contact metric (κ, μ)-manifold with κ < 1, we prove that it is flat or is locally isomorphic to the Lie group SU(2), SL(2), or E(1, 1) for n = 1 and that for n ⩾ 2 there are no weakly Einstein metrics on contact metric (κ, μ)-manifolds with 0 < κ < 1. For κ < 0, we get a classification of weakly Einstein contact metric (κ, μ)-manifolds. Finally, it is proved that a weakly Einstein almost cosymplectic (κ, μ)-manifold with κ < 0 is locally isomorphic to a solvable non-nilpotent Lie group.

GRAY CURVATURE IDENTITIES FOR ALMOST CONTACT METRIC MANIFOLDS

  • Mocanu, Raluca;Munteanu, Marian Ioan
    • Journal of the Korean Mathematical Society
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    • v.47 no.3
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    • pp.505-521
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    • 2010
  • Alfred Gray introduced in [8] three curvature identities for the class of almost Hermitian manifolds. Using the warped product construction and the Boothby-Wang fibration we will give an equivalent of these identities for the class of almost contact metric manifolds.

Effects of gender, age, and individual speakers on articulation rate in Seoul Korean spontaneous speech

  • Kim, Jungsun
    • Phonetics and Speech Sciences
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    • v.10 no.4
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    • pp.19-29
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    • 2018
  • The present study investigated whether there are differences in articulation rate by gender, age, and individual speakers in a spontaneous speech corpus produced by 40 Seoul Korean speakers. This study measured their articulation rates using a second-per-syllable metric and a syllable-per-second metric. The findings are as follows. First, in spontaneous Seoul Korean speech, there was a gender difference in articulation rates only in age group 10-19, among whom men tended to speak faster than women. Second, individual speakers showed variability in their rates of articulation. The tendency for some speakers to speak faster than others was variable. Finally, there were metric differences in articulation rate. That is, regarding the coefficients of variation, the values of the second-per-syllable metric were much higher than those for the syllable-per-second metric. The articulation rate for the syllable-per-second metric tended to be more distinct among individual speakers. The present results imply that data gathered in a corpus of Seoul Korean spontaneous speech may reflect speaker-specific differences in articulatory movements.

EXISTENCE OF HOMOTOPIC HARMONIC MAPS INTO METRIC SPACE OF NONPOSITIVE CURVATURE

  • Jeon, Myung-Jin
    • Communications of the Korean Mathematical Society
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    • v.10 no.4
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    • pp.931-941
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    • 1995
  • The definitions and techniques, which deals with homotopic harmonic maps from a compact Riemannian manifold into a compact metric space, developed by N. J. Korevaar and R. M. Schoen [7] can be applied to more general situations. In this paper, we prove that for a complicated domain, possibly noncompact Riemannian manifold with infinitely generated fundamental group, the existence of homotopic harmonic maps can be proved if the initial map is simple in some sense.

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