• Title/Summary/Keyword: Ricci-recurrent manifold

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ON QUASI RICCI SYMMETRIC MANIFOLDS

  • Kim, Jaeman
    • Korean Journal of Mathematics
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    • v.27 no.1
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    • pp.9-15
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    • 2019
  • In this paper, we study a type of Riemannian manifold, namely quasi Ricci symmetric manifold. Among others, we show that the scalar curvature of a quasi Ricci symmetric manifold is constant. In addition if the manifold is Einstein, then its Ricci tensor is zero. Also we prove that if the associated vector field of a quasi Ricci symmetric manifold is either recurrent or concurrent, then its Ricci tensor is zero.

ON GENERALIZED RICCI-RECURRENT TRANS-SASAKIAN MANIFOLDS

  • Kim, Jeong-Sik;Prasad, Rajendra;Tripathi, Mukut-Mani
    • Journal of the Korean Mathematical Society
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    • v.39 no.6
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    • pp.953-961
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    • 2002
  • Generalized Ricci-recurrent trans-Sasakian manifolds are studied. Among others, it is proved that a generalized Ricci-recurrent cosymplectic manifold is always recurrent Generalized Ricci-recurrent trans-Sasakian manifolds of dimension $\geq$ 5 are locally classified. It is also proved that if M is one of Sasakian, $\alpha$-Sasakian, Kenmotsu or $\beta$-Kenmotsu manifolds, which is gener-alized Ricci-recurrent with cyclic Ricci tensor and non-zero A (ξ) everywhere; then M is an Einstein manifold.

SOME NOTES ON NEARLY COSYMPLECTIC MANIFOLDS

  • Yildirim, Mustafa;Beyendi, Selahattin
    • Honam Mathematical Journal
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    • v.43 no.3
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    • pp.539-545
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    • 2021
  • In this paper, we study some symmetric and recurrent conditions of nearly cosymplectic manifolds. We prove that Ricci-semisymmetric and Ricci-recurrent nearly cosymplectic manifolds are Einstein and conformal flat nearly cosymplectic manifold is locally isometric to Riemannian product ℝ × N, where N is a nearly Kähler manifold.

ON GENERALIZED W3 RECURRENT RIEMANNIAN MANIFOLDS

  • Mohabbat Ali;Quddus Khan;Aziz Ullah Khan;Mohd Vasiulla
    • Honam Mathematical Journal
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    • v.45 no.2
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    • pp.325-339
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    • 2023
  • The object of the present work is to study a generalized W3 recurrent manifold. We obtain a necessary and sufficient condition for the scalar curvature to be constant in such a manifold. Also, sufficient condition for generalized W3 recurrent manifold to be special quasi-Einstein manifold are given. Ricci symmetric and decomposable generalized W3 recurrent manifold are studied. Finally, the existence of such a manifold is ensured by a non-trivial example.

On Generalized 𝜙-recurrent Kenmotsu Manifolds with respect to Quarter-symmetric Metric Connection

  • Hui, Shyamal Kumar;Lemence, Richard Santiago
    • Kyungpook Mathematical Journal
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    • v.58 no.2
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    • pp.347-359
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    • 2018
  • A Kenmotsu manifold $M^n({\phi},\;{\xi},\;{\eta},\;g)$, (n = 2m + 1 > 3) is called a generalized ${\phi}-recurrent$ if its curvature tensor R satisfies $${\phi}^2(({\nabla}_wR)(X,Y)Z)=A(W)R(X,Y)Z+B(W)G(X,Y)Z$$ for all $X,\;Y,\;Z,\;W{\in}{\chi}(M)$, where ${\nabla}$ denotes the operator of covariant differentiation with respect to the metric g, i.e. ${\nabla}$ is the Riemannian connection, A, B are non-vanishing 1-forms and G is given by G(X, Y)Z = g(Y, Z)X - g(X, Z)Y. In particular, if A = 0 = B then the manifold is called a ${\phi}-symmetric$. Now, a Kenmotsu manifold $M^n({\phi},\;{\xi},\;{\eta},\;g)$, (n = 2m + 1 > 3) is said to be generalized ${\phi}-Ricci$ recurrent if it satisfies $${\phi}^2(({\nabla}_wQ)(Y))=A(X)QY+B(X)Y$$ for any vector field $X,\;Y{\in}{\chi}(M)$, where Q is the Ricci operator, i.e., g(QX, Y) = S(X, Y) for all X, Y. In this paper, we study generalized ${\phi}-recurrent$ and generalized ${\phi}-Ricci$ recurrent Kenmotsu manifolds with respect to quarter-symmetric metric connection and obtain a necessary and sufficient condition of a generalized ${\phi}-recurrent$ Kenmotsu manifold with respect to quarter symmetric metric connection to be generalized Ricci recurrent Kenmotsu manifold with respect to quarter symmetric metric connection.

On Quasi-Conformally Recurrent Manifolds with Harmonic Quasi-Conformal Curvature Tensor

  • Shaikh, Absos Ali;Roy, Indranil
    • Kyungpook Mathematical Journal
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    • v.51 no.1
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    • pp.109-124
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    • 2011
  • The main objective of the paper is to provide a full classification of quasi-conformally recurrent Riemannian manifolds with harmonic quasi-conformal curvature tensor. Among others it is shown that a quasi-conformally recurrent manifold with harmonic quasi-conformal curvature tensor is any one of the following: (i) quasi-conformally symmetric, (ii) conformally flat, (iii) manifold of constant curvature, (iv) vanishing scalar curvature, (v) Ricci recurrent.

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.

A TYPE OF WEAKLY SYMMETRIC STRUCTURE ON A RIEMANNIAN MANIFOLD

  • Kim, Jaeman
    • Korean Journal of Mathematics
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    • v.30 no.1
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    • pp.61-66
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    • 2022
  • A new type of Riemannian manifold called semirecurrent manifold has been defined and some of its geometric properties are studied. Among others we show that the scalar curvature of semirecurrent manifold is constant and hence semirecurrent manifold is also concircularly recurrent. In addition, we show that the associated 1-form (resp. the associated vector field) of semirecurrent manifold is closed (resp. an eigenvector of its Ricci tensor). Furthermore, we prove that if a Riemannian product manifold is semirecurrent, then either one decomposition manifold is locally symmetric or the other decomposition manifold is a space of constant curvature.