• Title, Summary, Keyword: Riemannian manifold

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SUBMANIFOLDS OF AN ALMOST r-PARACONTACT RIEMANNIAN MANIFOLD ENDOWED WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION

  • Ahmad, Mobin;Jun, Jae-Bok
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.4
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    • pp.653-665
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    • 2009
  • We define a semi-symmetric non-metric connection in an almost r-paracontact Riemannian manifold and we consider submanifolds of an almost r-paracontact Riemannian manifold endowed with a semi-symmetric non-metric connection and obtain Gauss and Codazzi equations, Weingarten equation and curvature tensor for submanifolds of an almost r-paracontact Riemannian manifold endowed with a semi-symmetric non-metric connection.

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SUBMANIFOLDS OF AN ALMOST r-PARACONTACT RIEMANNIAN MANIFOLD ENDOWED WITH A SEMI-SYMMETRIC METRIC CONNECTION

  • Ahmad, Mobin;Jun, Jae-Bok
    • Honam Mathematical Journal
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    • v.32 no.3
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    • pp.363-374
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    • 2010
  • We define a semi-symmetric metric connection in an almost r-paracontact Riemannian manifold and we consider submanifolds of an almost r-paracontact Riemannian manifold endowed with a semi-symmetric metric connection and obtain Gauss and Codazzi equations, Weingarten equation and curvature tensor for submanifolds of an almost r-paracontact Riemannian manifold endowed with a semi-symmetric metric connection.

ON ALMOST r-PARACONTACT RIEMANNIAN MANIFOLD WITH A CERTAIN CONNECTION

  • Ahmad, Mobin;Haseeb, Abdul;Jun, Jae-Bok;Rahman, Shamsur
    • Communications of the Korean Mathematical Society
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    • v.25 no.2
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    • pp.235-243
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    • 2010
  • In a Riemannian manifold, the existence of a new connection is proved. In particular cases, this connection reduces to several symmetric, semi-symmetric and quarter symmetric connections, even some of them are not introduced so far. So, in this paper, we define a quarter symmetric semi-metric connection in an almost r-paracontact Riemannian manifold and consider invariant, non-invariant and anti-invariant hypersurfaces of an almost r-paracontact Riemannian manifold with that connection.

GCR-LIGHTLIKE SUBMANIFOLDS OF A SEMI-RIEMANNIAN PRODUCT MANIFOLD

  • Kumar, Sangeet;Kumar, Rakesh;Nagaich, Rakesh Kumar
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.883-899
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    • 2014
  • We introduce GCR-lightlike submanifold of a semi-Riemannian product manifold and give an example. We study geodesic GCR-lightlike submanifolds of a semi-Riemannian product manifold and obtain some necessary and sufficient conditions for a GCR-lightlike submanifold to be a GCR-lightlike product. Finally, we discuss minimal GCR-lightlike submanifolds of a semi-Riemannian product manifold.

CURVATURES ON THE ABBENA-THURSTON MANIFOLD

  • Han, Ju-Wan;Kim, Hyun Woong;Pyo, Yong-Soo
    • Honam Mathematical Journal
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    • v.38 no.2
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    • pp.359-366
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    • 2016
  • Let H be the 3-dimensional Heisenberg group, ($G=H{\times}S^1$, g) a product Riemannian manifold of Riemannian manifolds H and S with arbitrarily given left invariant Riemannian metrics respectively, and ${\Gamma}$ the discrete subgroup of G with integer entries. Then, on the Riemannian manifold ($M:=G/{\Gamma}$, ${\Pi}^*g=\bar{g}$), ${\Pi}:G{\rightarrow}G/{\Gamma}$, we evaluate the scalar curvature and the Ricci curvature.

REMARKS ON METALLIC MAPS BETWEEN METALLIC RIEMANNIAN MANIFOLDS AND CONSTANCY OF CERTAIN MAPS

  • Akyol, Mehmet Akif
    • Honam Mathematical Journal
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    • v.41 no.2
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    • pp.343-356
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    • 2019
  • In this paper, we introduce metallic maps between metallic Riemannian manifolds, provide an example and obtain certain conditions for such maps to be totally geodesic. We also give a sufficient condition for a map between metallic Riemannian manifolds to be harmonic map. Then we investigate the constancy of certain maps between metallic Riemannian manifolds and various manifolds by imposing the holomorphic-like condition. Moreover, we check the reverse case and show that some such maps are constant if there is a condition for this.

EMBEDDING OPEN RIEMANN SURFACES IN 4-DIMENSIONAL RIEMANNIAN MANIFOLDS

  • Ko, Seokku
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.1
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    • pp.205-214
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    • 2016
  • Any open Riemann surface has a conformal model in any orientable Riemannian manifold of dimension 4. Precisely, we will prove that, given any open Riemann surface, there is a conformally equivalent model in a prespecified orientable 4-dimensional Riemannian manifold. This result along with [5] now shows that an open Riemann surface admits conformal models in any Riemannian manifold of dimension ${\geq}3$.

SUBMANIFOLDS OF AN ALMOST r-PARACONTACT RIEMANNIAN MANIFOLD ENDOWED WITH A QUARTER-SYMMETRIC NON-METRIC CONNECTION

  • Ahmad, Mobin;Jun, Jae-Bok;Haseeb, Abdul
    • Journal of the Chungcheong Mathematical Society
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    • v.24 no.1
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    • pp.91-104
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    • 2011
  • We define a quarter-symmetric non-metric connection in an almost r-paracontact Riemannian manifold and we consider the submanifolds of an almost r-paracontact Riemannian manifold endowed with a quarter-symmetric non-metric connection. We also obtain the Gauss, Codazzi and Weingarten equations and the curvature tensor for the submanifolds of an almost r-paracontact Riemannian manifold endowed with a quarter-symmetric non-metric connection.

Critical rimennian metrics on cosymplectic manifolds

  • Kim, Byung-Hak
    • Journal of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.553-562
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    • 1995
  • In a Recent paper [3], D. Chinea, M. Delon and J. C. Marrero proved that a cosymplectic manifold is formal and constructed an example of compact cosymplectic manifold which is not a global product of a Kaehler manifold with the circle. In this paper we study the compact cosymplectic manifolds with critical Riemannian metrics.

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ON KENMOTSU MANIFOLDS

  • JUN JAE-BOK;DE UDAY CHAND;PATHAK GOUTAM
    • Journal of the Korean Mathematical Society
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    • v.42 no.3
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    • pp.435-445
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    • 2005
  • The purpose of this paper is to study a Kenmotsu manifold which is derived from the almost contact Riemannian manifold with some special conditions. In general, we have some relations about semi-symmetric, Ricci semi-symmetric or Weyl semisymmetric conditions in Riemannian manifolds. In this paper, we partially classify the Kenmotsu manifold and consider the manifold admitting a transformation which keeps Riemannian curvature tensor and Ricci tensor invariant.