• Title, Summary, Keyword: ${\eta}$-Einstein manifolds

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ON A CLASS OF THREE-DIMENSIONAL TRANS-SASAKIAN MANIFOLDS

  • De, Uday Chand;De, Krishnendu
    • Communications of the Korean Mathematical Society
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    • v.27 no.4
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    • pp.795-808
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    • 2012
  • The object of the present paper is to study 3-dimensional trans-Sasakian manifolds with conservative curvature tensor and also 3-dimensional conformally flat trans-Sasakian manifolds. Next we consider compact connected $\eta$-Einstein 3-dimensional trans-Sasakian manifolds. Finally, an example of a 3-dimensional trans-Sasakian manifold is given, which verifies our results.

Symmetry Properties of 3-dimensional D'Atri Spaces

  • Belkhelfa, Mohamed;Deszcz, Ryszard;Verstraelen, Leopold
    • Kyungpook Mathematical Journal
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    • v.46 no.3
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    • pp.367-376
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    • 2006
  • We investigate semi-symmetry and pseudo-symmetry of some 3-dimensional Riemannian manifolds: the D'Atri spaces, the Thurston geometries as well as the ${\eta}$-Einstein manifolds. We prove that all these manifolds are pseudo-symmetric and that many of them are not semi-symmetric.

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η-Ricci Solitons in δ-Lorentzian Trans Sasakian Manifolds with a Semi-symmetric Metric Connection

  • Siddiqi, Mohd Danish
    • Kyungpook Mathematical Journal
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    • v.59 no.3
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    • pp.537-562
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    • 2019
  • The aim of the present paper is to study the ${\delta}$-Lorentzian trans-Sasakian manifold endowed with semi-symmetric metric connections admitting ${\eta}$-Ricci Solitons and Ricci Solitons. We find expressions for the curvature tensor, the Ricci curvature tensor and the scalar curvature tensor of ${\delta}$-Lorentzian trans-Sasakian manifolds with a semisymmetric-metric connection. Also, we discuses some results on quasi-projectively flat and ${\phi}$-projectively flat manifolds endowed with a semi-symmetric-metric connection. It is shown that the manifold satisfying ${\bar{R}}.{\bar{S}}=0$, ${\bar{P}}.{\bar{S}}=0$ is an ${\eta}$-Einstein manifold. Moreover, we obtain the conditions for the ${\delta}$-Lorentzian trans-Sasakian manifolds with a semisymmetric-metric connection to be conformally flat and ${\xi}$-conformally flat.

A CLASSIFICATION OF (κ, μ)-CONTACT METRIC MANIFOLDS

  • Yildiz, Ahmet;De, Uday Chand
    • Communications of the Korean Mathematical Society
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    • v.27 no.2
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    • pp.327-339
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    • 2012
  • In this paper we study $h$-projectively semisymmetric, ${\phi}$-pro-jectively semisymmetric, $h$-Weyl semisymmetric and ${\phi}$-Weyl semisym- metric non-Sasakian ($k$, ${\mu}$)-contact metric manifolds. In all the cases the manifold becomes an ${\eta}$-Einstein manifold. As a consequence of these results we obtain that if a 3-dimensional non-Sasakian ($k$, ${\mu}$)-contact metric manifold satisfies such curvature conditions, then the manifold reduces to an N($k$)-contact metric manifold.

Some Symmetric Properties on (LCS)n-manifolds

  • Venkatesha, Venkatesha;Naveen Kumar, Rahuthanahalli Thimmegowda
    • Kyungpook Mathematical Journal
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    • v.55 no.1
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    • pp.149-156
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    • 2015
  • We analyze the $(LCS)_n$-manifolds endowed with some symmetric properties, focusing on Ricci tensor and the 1-form ${\gamma}$. We study some properties of special Weakly Ricci-Symmetric $(LCS)_n$-manifolds and also shown that Weakly ${\phi}$-Ricci Symmetric $(LCS)_n$-manifold is an ${\eta}$-Einstein manifold.

SOME RECURRENT PROPERTIES OF LP-SASAKIAN NANIFOLDS

  • Venkatesha, Venkatesha;Somashekhara., P.
    • Korean Journal of Mathematics
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    • v.27 no.3
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    • pp.793-801
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    • 2019
  • The aim of the present paper is to study certain recurrent properties of LP-Sasakian manifolds. Here we first describe Ricci ${\eta}$-recurrent LP-Sasakian manifolds. Further we study semi-generalized recurrent and three dimensional locally generalized concircularly ${\phi}$-recurrent LP-Sasakian manifolds and got interesting results.

ON Φ-RECURRENT (k, μ)-CONTACT METRIC MANIFOLDS

  • Jun, Jae-Bok;Yildiz, Ahmet;De, Uday Chand
    • Bulletin of the Korean Mathematical Society
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    • v.45 no.4
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    • pp.689-700
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    • 2008
  • In this paper we prove that a $\phi$-recurrent (k, $\mu$)-contact metric manifold is an $\eta$-Einstein manifold with constant coefficients. Next, we prove that a three-dimensional locally $\phi$-recurrent (k, $\mu$)-contact metric manifold is the space of constant curvature. The existence of $\phi$-recurrent (k, $\mu$)-manifold is proved by a non-trivial example.

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.