• Title/Summary/Keyword: 4-Manifold

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THREE-DIMENSIONAL LORENTZIAN PARA-KENMOTSU MANIFOLDS AND YAMABE SOLITONS

  • Pankaj, Pankaj;Chaubey, Sudhakar K.;Prasad, Rajendra
    • Honam Mathematical Journal
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    • v.43 no.4
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    • pp.613-626
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    • 2021
  • The aim of the present work is to study the properties of three-dimensional Lorentzian para-Kenmotsu manifolds equipped with a Yamabe soliton. It is proved that every three-dimensional Lorentzian para-Kenmotsu manifold is Ricci semi-symmetric if and only if it is Einstein. Also, if the metric of a three-dimensional semi-symmetric Lorentzian para-Kenmotsu manifold is a Yamabe soliton, then the soliton is shrinking and the flow vector field is Killing. We also study the properties of three-dimensional Ricci symmetric and 𝜂-parallel Lorentzian para-Kenmotsu manifolds with Yamabe solitons. Finally, we give a non-trivial example of three-dimensional Lorentzian para-Kenmotsu manifold.

Some Geometric Properties of η-Ricci Solitons on α-Lorentzian Sasakian Manifolds

  • Shashikant, Pandey;Abhishek, Singh;Rajendra, Prasad
    • Kyungpook Mathematical Journal
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    • v.62 no.4
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    • pp.737-749
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    • 2022
  • We investigate the geometric properties of 𝜂*-Ricci solitons on α-Lorentzian Sasakian (α-LS) manifolds, and show that a Ricci semisymmetric 𝜂*-Ricci soliton on an α-LS manifold is an 𝜂*-Einstein manifold. Further, we study 𝜑*-symmetric 𝜂*-Ricci solitons on such manifolds. We prove that 𝜑*-Ricci symmetric 𝜂*-Ricci solitons on an α-LS manifold are also 𝜂*-Einstein manifolds and provide an example of a 3-dimensional α-LS manifold for the existence of such solitons.

The Geometry of 𝛿-Ricci-Yamabe Almost Solitons on Paracontact Metric Manifolds

  • Somnath Mondal;Santu Dey;Young Jin Suh;Arindam Bhattacharyya
    • Kyungpook Mathematical Journal
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    • v.63 no.4
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    • pp.623-638
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    • 2023
  • In this article we study a 𝛿-Ricci-Yamabe almost soliton within the framework of paracontact metric manifolds. In particular we study 𝛿-Ricci-Yamabe almost soliton and gradient 𝛿-Ricci-Yamabe almost soliton on K-paracontact and para-Sasakian manifolds. We prove that if a K-paracontact metric g represents a 𝛿-Ricci-Yamabe almost soliton with the non-zero potential vector field V parallel to 𝜉, then g is Einstein with Einstein constant -2n. We also show that there are no para-Sasakian manifolds that admit a gradient 𝛿-Ricci-Yamabe almost soliton. We demonstrate a 𝛿-Ricci-Yamabe almost soliton on a (𝜅, 𝜇)-paracontact manifold.

SCREEN GENERIC LIGHTLIKE SUBMERSIONS

  • Gaurav Sharma;Sangeet Kumar;Dinesh Kumar Sharma
    • Honam Mathematical Journal
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    • v.45 no.4
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    • pp.629-647
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    • 2023
  • We introduce the study of a new class of a lightlike submersion d. Then, we derive a relationship between the holomorphic section 𝜙 : K1 → K' from a screen generic lightlike submanifold of an indefinite Kaehler manifold K2 onto an indefinite almost Hermitian manifold K', and show that for this case K' must be an indefinite Kaehler manifold. Then, we derive a relationship between the holomorphic sectional curvatures of K2 and K'. Finally, we present a classification theorem for a screen generic lightlike submersion, giving the relationship between the sectional curvatures of the total space K2 and the fibers.

CLASSIFICATION OF TWISTED PRODUCT LIGHTLIKE SUBMANIFOLDS

  • Sangeet Kumar;Megha Pruthi
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.4
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    • pp.1003-1016
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    • 2023
  • In this paper, we introduce the idea of twisted product lightlike submanifolds of semi-Riemannian manifolds and provide non-trivial examples of such lightlike submanifolds. Then, we prove the non-existence of proper isotropic or totally lightlike twisted product submanifolds of a semi-Riemannian manifold. We also show that for a twisted product lightlike submanifold of a semi-Riemannian manifold, the induced connection ∇ is not a metric connection. Further, we prove that a totally umbilical SCR-lightlike submanifold of an indefinite Kaehler manifold ${\tilde{M}}$ does not admit any twisted product SCR-lightlike submanifold of the type M×ϕMT, where M is a totally real submanifold and MT is a holomorphic submanifold of ${\tilde{M}}$. Consequently, we obtain a geometric inequality for the second fundamental form of twisted product SCR-lightlike submanifolds of the type MT×ϕM of an indefinite Kaehler manifold ${\tilde{M}}$, in terms of the gradient of ln ϕ, where ϕ stands for the twisting function. Subsequently, the equality case of this inequality is discussed. Finally, we construct a non-trivial example of a twisted product SCR-lightlike submanifold in an indefinite Kaehler manifold.

ON ACTION SPECTRUM BUNDLE

  • Cho, Yong-Seung;Yoon, Jin-Yue
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.4
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    • pp.741-751
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    • 2001
  • In this paper when $(M, \omega)$ is a compact weakly exact symplectic manifold with nonempty boundary satisfying $c_1|{\pi}_2(M)$ = 0, we construct an action spectrum bundle over the group of Hamil-tonian diffeomorphisms of the manifold M generated by the time-dependent Hamiltonian vector fields, whose fibre is nowhere dense and invariant under symplectic conjugation.

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