• Title, Summary, Keyword: invariant metric

Search Result 72, Processing Time 0.037 seconds

DIFFERENTIAL GEOMETRIC PROPERTIES ON THE HEISENBERG GROUP

  • Park, Joon-Sik
    • Journal of the Korean Mathematical Society
    • /
    • v.53 no.5
    • /
    • pp.1149-1165
    • /
    • 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.

ON SEMI-INVARIANT SUBMANIFOLDS OF A NEARLY KENMOTSU MANIFOLD WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION

  • Ahmad, Mobin;Jun, Jae-Bok
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.23 no.2
    • /
    • pp.257-266
    • /
    • 2010
  • We define a semi-symmetric non-metric connection in a nearly Kenmotsu manifold and we study semi-invariant submanifolds of a nearly Kenmotsu manifold endowed with a semi-symmetric non-metric connection. Moreover, we discuss the integrability of distributions on semi-invariant submanifolds of a nearly Kenmotsu manifold with a semi-symmetric non-metric connection.

ON SEMI-INVARIANT SUBMANIFOLDS OF A NEARLY KENMOTSU MANIFOLD WITH A QUARTER SYMMETRIC NON-METRIC CONNECTION

  • Ahmad, Mobin;Jun, Jae-Bok
    • The Pure and Applied Mathematics
    • /
    • v.18 no.1
    • /
    • pp.1-11
    • /
    • 2011
  • We define a quarter symmetric non-metric connection in a nearly Ken-motsu manifold and we study semi-invariant submanifolds of a nearly Kenmotsu manifold endowed with a quarter symmetric non-metric connection. Moreover, we discuss the integrability of the distributions on semi-invariant submanifolds of a nearly Kenmotsu manifold with a quarter symmetric non-metric connection.

HARMONIC HOMOMORPHISMS BETWEEN TWO LIE GROUPS

  • Son, Heui-Sang;Kim, Hyun Woong;Park, Joon-Sik
    • Honam Mathematical Journal
    • /
    • v.38 no.1
    • /
    • pp.1-8
    • /
    • 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.

THE TRANSFORMATION GROUPS AND THE ISOMETRY GROUPS

  • Kim, Young-Wook
    • Bulletin of the Korean Mathematical Society
    • /
    • v.26 no.1
    • /
    • pp.47-52
    • /
    • 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.

  • PDF

Hypersurfaces of an almost r-paracontact Riemannian Manifold Endowed with a Quarter Symmetric Non-metric Connection

  • Ahmad, Mobin;Haseeb, Abdul;Ozgur, Cihan
    • Kyungpook Mathematical Journal
    • /
    • v.49 no.3
    • /
    • pp.533-543
    • /
    • 2009
  • We define a quarter symmetric non-metric connection in an almost r-paracontact Riemannian manifold and we consider invariant, non-invariant and anti-invariant hypersurfaces of an almost r-paracontact Riemannian manifold endowed with a quarter symmetric non-metric connection.

HYPERSURFACES OF ALMOST γ-PARACONTACT RIEMANNIAN MANIFOLD ENDOWED WITH SEMI-SYMMETRIC METRIC CONNECTION

  • Jun, Jae-Bok;Ahmad, Mobin
    • Bulletin of the Korean Mathematical Society
    • /
    • v.46 no.5
    • /
    • pp.895-903
    • /
    • 2009
  • We define a semi-symmetric metric connection in an almost $\gamma$-paracontact Riemannian manifold and we consider invariant, non-invariant and anti-invariant hypersurfaces of an almost $\gamma$-paracontact Riemannian manifold endowed with a semi-symmetric metric connection.

COMMON FIXED POINT THEOREM AND INVARIANT APPROXIMATION IN COMPLETE LINEAR METRIC SPACES

  • Nashine, Hemant Kumar
    • East Asian mathematical journal
    • /
    • v.28 no.5
    • /
    • pp.533-541
    • /
    • 2012
  • A common fixed point result of Gregus type for subcompatible mappings defined on a complete linear metric space is obtained. The considered underlying space is generalized from Banach space to complete linear metric spaces, which include Banach space and complete metrizable locally convex spaces. Invariant approximation results have also been determined as its application.

YANG-MILLS CONNECTIONS ON CLOSED LIE GROUPS

  • Pyo, Yong-Soo;Shin, Young-Lim;Park, Joon-Sik
    • Honam Mathematical Journal
    • /
    • v.32 no.4
    • /
    • pp.651-661
    • /
    • 2010
  • In this paper, we obtain a necessary and sufficient condition for a left invariant connection in the tangent bundle over a closed Lie group with a left invariant metric to be a Yang-Mills connection. Moreover, we have a necessary and sufficient condition for a left invariant connection with a torsion-free Weyl structure in the tangent bundle over SU(2) with a left invariant Riemannian metric g to be a Yang-Mills connection.

HYPERSURFACES OF ALMOST γ-PARACONTACT RIEMANNIAN MANIFOLD ENDOWED WITH A QUARTER SYMMETRIC METRIC CONNECTION

  • Ahmad, Mobin;Jun, Jae-Bok;Haseeb, Abdul
    • Bulletin of the Korean Mathematical Society
    • /
    • v.46 no.3
    • /
    • pp.477-487
    • /
    • 2009
  • We define a quarter symmetric metric connection in an almost r-paracontact Riemannian manifold and we consider invariant, noninvariant and anti-invariant hypersurfaces of an almost r-paracontact Riemannian manifold endowed with a quarter symmetric metric connection.