• Title/Summary/Keyword: Ricci-recurrent

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Conformally Flat Totally Umbilical Submanifolds in Some Semi-Riemannian Manifolds

  • Ewert-Krzemieniewski, Stanislaw
    • Kyungpook Mathematical Journal
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    • v.48 no.2
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    • pp.183-194
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    • 2008
  • We prove that totally umbilical submanifold M of an extended quasi-recurren manifold is also extended quasi-recurrent. If, moreover, M is conformally flat then, locally, M is isometric to the manifold with known metric. Some curvature properties of such submanifold are investigated. Making use of these results we shall prove the existence of totally umbilical submanifold being pseudosymmetric in the sense of Ryszard Deszcz and satisfying some other curvature conditions.

NEARLY SASAKIAN MANIFOLD SATISFYING

  • Kim, Chong-Hon;Kim, Byong-Du
    • Bulletin of the Korean Mathematical Society
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    • v.21 no.1
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    • pp.21-26
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    • 1984
  • The notion of nearly Sasakian manifold was introduced in [1] and Z-Olszak has studied certain properties in [2] and [3]. In section (2) of this paper, we show that a nearly Sasakian manifold M admitting .GAMMA.$_{ji}$ $^{h}$ such that .del.$_{1}$ $R_{kji}$$^{h}$ =0 is of contant scalar curvature and the covariant derivate of the Ricci tensor of M is a symmetric tensor. In the last section, we shall deal with a recurrent and conformal recurrent nearly Sasakian manifold.d.

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SPACETIMES ADMITTING DIVERGENCE FREE m-PROJECTIVE CURVATURE TENSOR

  • Uday Chand De;Dipankar Hazra
    • Communications of the Korean Mathematical Society
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    • v.39 no.1
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    • pp.201-210
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    • 2024
  • This paper is concerned with the study of spacetimes satisfying div 𝓜 = 0, where "div" denotes the divergence and 𝓜 is the m-projective curvature tensor. We establish that a perfect fluid spacetime with div 𝓜 = 0 is a generalized Robertson-Walker spacetime and vorticity free; whereas a four-dimensional perfect fluid spacetime becomes a Robertson-Walker spacetime. Moreover, we establish that a Ricci recurrent spacetime with div 𝓜 = 0 represents a generalized Robertson-Walker spacetime.