• Title/Summary/Keyword: Codazzi type

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REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WHOSE SHAPE OPERATOR IS OF CODAZZI TYPE IN GENERALIZED TANAKA-WEBSTER CONNECTION

  • Cho, Kyusuk;Lee, Hyunjin;Pak, Eunmi
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.1
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    • pp.57-68
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    • 2015
  • In this paper, we give a non-existence theorem of Hopf hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$, $m{\geq}3$, whose shape operator is of Codazzi type in generalized Tanaka-Webster connection $\hat{\nabla}^{(k)}$.

𝜂-RICCI SOLITONS ON PARA-KENMOTSU MANIFOLDS WITH SOME CURVATURE CONDITIONS

  • Mondal, Ashis
    • Korean Journal of Mathematics
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    • v.29 no.4
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    • pp.705-714
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    • 2021
  • In the present paper, we study 𝜂-Ricci solitons on para-Kenmotsu manifolds with Codazzi type of the Ricci tensor. We study 𝜂-Ricci solitons on para-Kenmotsu manifolds with cyclic parallel Ricci tensor. We also study 𝜂-Ricci solitons on 𝜑-conformally semi-symmetric, 𝜑-Ricci symmetric and conformally Ricci semi-symmetric para-Kenmotsu manifolds. Finally, we construct an example of a three-dimensional para-Kenmotsu manifold which admits 𝜂-Ricci solitons.

On a Classification of Almost Kenmotsu Manifolds with Generalized (k, µ)'-nullity Distribution

  • Ghosh, Gopal;Majhi, Pradip;Chand De, Uday
    • Kyungpook Mathematical Journal
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    • v.58 no.1
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    • pp.137-148
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    • 2018
  • In the present paper we prove that in an almost Kenmotsu manifold with generalized $(k,{\mu})^{\prime}-nullity$ distribution the three conditions: (i) the Ricci tensor of $M^{2n+1}$ is of Codazzi type, (ii) the manifold $M^{2n+1}$ satisfies div C = 0, (iii) the manifold $M^{2n+1}$ is locally isometric to $H^{n+1}(-4){\times}R^n$, are equivalent. Also we prove that if the manifold satisfies the cyclic parallel Ricci tensor, then the manifold is locally isometric to $H^{n+1}(-4){\times}\mathbb{R}^n$.

SASAKIAN 3-MANIFOLDS SATISFYING SOME CURVATURE CONDITIONS ASSOCIATED TO Ƶ-TENSOR

  • Dey, Dibakar;Majhi, Pradip
    • The Pure and Applied Mathematics
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    • v.28 no.2
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    • pp.143-153
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    • 2021
  • In this paper, we study some curvature properties of Sasakian 3-manifolds associated to Ƶ-tensor. It is proved that if a Sasakian 3-manifold (M, g) satisfies one of the conditions (1) the Ƶ-tensor is of Codazzi type, (2) M is Ƶ-semisymmetric, (3) M satisfies Q(Ƶ, R) = 0, (4) M is projectively Ƶ-semisymmetric, (5) M is Ƶ-recurrent, then (M, g) is of constant curvature 1. Several consequences are drawn from these results.

STUDY OF P-CURVATURE TENSOR IN THE SPACE-TIME OF GENERAL RELATIVITY

  • Ganesh Prasad Pokhariyal;Sudhakar Kumar Chaubey
    • Honam Mathematical Journal
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    • v.45 no.2
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    • pp.316-324
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    • 2023
  • The P-curvature tensor has been studied in the space-time of general relativity and it is found that the contracted part of this tensor vanishes in the Einstein space. It is shown that Rainich conditions for the existence of non-null electro variance can be obtained by P𝛼𝛽. It is established that the divergence of tensor G𝛼𝛽 defined with the help of P𝛼𝛽 and scalar P is zero, so that it can be used to represent Einstein field equations. It is shown that for V4 satisfying Einstein like field equations, the tensor P𝛼𝛽 is conserved, if the energy momentum tensor is Codazzi type. The space-time satisfying Einstein's field equations with vanishing of P-curvature tensor have been considered and existence of Killing, conformal Killing vector fields and perfect fluid space-time has been established.

ON ALMOST QUASI RICCI SYMMETRIC MANIFOLDS

  • Kim, Jaeman
    • Communications of the Korean Mathematical Society
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    • v.35 no.2
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    • pp.603-611
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    • 2020
  • The purpose of this note is to introduce a type of Riemannian manifold called an almost quasi Ricci symmetric manifold and investigate the several properties of such a manifold on which some geometric conditions are imposed. And the existence of such a manifold is ensured by a proper example.

ON SOME CLASSES OF WEAKLY Z-SYMMETRIC MANIFOLDS

  • Lalnunsiami, Kingbawl;Singh, Jay Prakash
    • Communications of the Korean Mathematical Society
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    • v.35 no.3
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    • pp.935-951
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    • 2020
  • The aim of the paper is to study some geometric properties of weakly Z-symmetric manifolds. Weakly Z-symmetric manifolds with Codazzi type and cyclic parallel Z tensor are studied. We consider Einstein weakly Z-symmetric manifolds and conformally flat weakly Z-symmetric manifolds. Next, it is shown that a totally umbilical hypersurface of a conformally flat weakly Z-symmetric manifolds is of quasi constant curvature. Also, decomposable weakly Z-symmetric manifolds are studied and some examples are constructed to support the existence of such manifolds.

On Generalized Ricci Recurrent Spacetimes

  • Dey, Chiranjib
    • Kyungpook Mathematical Journal
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    • v.60 no.3
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    • pp.571-584
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    • 2020
  • The object of the present paper is to characterize generalized Ricci recurrent (GR4) spacetimes. Among others things, it is proved that a conformally flat GR4 spacetime is a perfect fluid spacetime. We also prove that a GR4 spacetime with a Codazzi type Ricci tensor is a generalized Robertson Walker spacetime with Einstein fiber. We further show that in a GR4 spacetime with constant scalar curvature the energy momentum tensor is semisymmetric. Further, we obtain several corollaries. Finally, we cite some examples which are sufficient to demonstrate that the GR4 spacetime is non-empty and a GR4 spacetime is not a trivial case.

The 𝒲-curvature Tensor on Relativistic Space-times

  • Abu-Donia, Hassan;Shenawy, Sameh;Syied, Abdallah Abdelhameed
    • Kyungpook Mathematical Journal
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    • v.60 no.1
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    • pp.185-195
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    • 2020
  • This paper aims to study the 𝒲-curvature tensor on relativistic space-times. The energy-momentum tensor T of a space-time having a semi-symmetric 𝒲-curvature tensor is semi-symmetric, whereas the whereas the energy-momentum tensor T of a space-time having a divergence free 𝒲-curvature tensor is of Codazzi type. A space-time having a traceless 𝒲-curvature tensor is Einstein. A 𝒲-curvature flat space-time is Einstein. Perfect fluid space-times which admits 𝒲-curvature tensor are considered.