• Title/Summary/Keyword: almost contact metric manifolds

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NEARLY KAEHLERIAN PRODUCT MANIFOLDS OF TWO ALMOST CONTACT METRIC MANIFOLDS

  • Ki, U-Hang;Kim, In-Bae;Lee, Eui-Won
    • Bulletin of the Korean Mathematical Society
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    • v.21 no.2
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    • pp.61-66
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    • 1984
  • It is well-known that the most interesting non-integrable almost Hermitian manifold are the nearly Kaehlerian manifolds ([2] and [3]), and that there exists a complex but not a Kaehlerian structure on Riemannian product manifolds of two normal contact manifolds [4]. The purpose of the present paper is to study nearly Kaehlerian product manifolds of two almost contact metric manifolds and investigate the geometrical structures of these manifolds. Unless otherwise stated, we shall always assume that manifolds and quantities are differentiable of class $C^{\infty}$. In Paragraph 1, we give brief discussions of almost contact metric manifolds and their Riemannian product manifolds. In paragraph 2, we investigate the perfect conditions for Riemannian product manifolds of two almost contact metric manifolds to be nearly Kaehlerian and the non-existence of a nearly Kaehlerian product manifold of contact metric manifolds. Paragraph 3 will be devoted to a proof of the following; A conformally flat compact nearly Kaehlerian product manifold of two almost contact metric manifolds is isomatric to a Riemannian product manifold of a complex projective space and a flat Kaehlerian manifold..

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A NEW TYPE WARPED PRODUCT METRIC IN CONTACT GEOMETRY

  • Mollaogullari, Ahmet;Camci, Cetin
    • Honam Mathematical Journal
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    • v.44 no.1
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    • pp.62-77
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    • 2022
  • This study presents an 𝛼-Sasakian structure on the product manifold M1 × 𝛽(I), where M1 is a Kähler manifold with an exact 1-form, and 𝛽(I) is an open curve. It then defines a new type warped product metric to study the warped product of almost Hermitian manifolds with almost contact metric manifolds, contact metric manifolds, and K-contact manifolds.

GRAY CURVATURE IDENTITIES FOR ALMOST CONTACT METRIC MANIFOLDS

  • Mocanu, Raluca;Munteanu, Marian Ioan
    • Journal of the Korean Mathematical Society
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    • v.47 no.3
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    • pp.505-521
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    • 2010
  • Alfred Gray introduced in [8] three curvature identities for the class of almost Hermitian manifolds. Using the warped product construction and the Boothby-Wang fibration we will give an equivalent of these identities for the class of almost contact metric manifolds.

GOLDEN PARA-CONTACT METRIC MANIFOLDS

  • Beldjilali, Gherici;Bouzir, Habib
    • Communications of the Korean Mathematical Society
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    • v.37 no.4
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    • pp.1209-1219
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    • 2022
  • The purpose of the present paper is to introduce a new class of almost para-contact metric manifolds namely, Golden para-contact metric manifolds. Then, we are particularly interested in a more special type called Golden para-Sasakian manifolds, where we will study their fundamental properties and we present many examples which justify their study.

ON 3-DIMENSIONAL NORMAL ALMOST CONTACT METRIC MANIFOLDS SATISFYING CERTAIN CURVATURE CONDITIONS

  • De, Uday Chand;Mondal, Abul Kalam
    • Communications of the Korean Mathematical Society
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    • v.24 no.2
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    • pp.265-275
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    • 2009
  • The object of the present paper is to study 3-dimensional normal almost contact metric manifolds satisfying certain curvature conditions. Among others it is proved that a parallel symmetric (0, 2) tensor field in a 3-dimensional non-cosympletic normal almost contact metric manifold is a constant multiple of the associated metric tensor and there does not exist a non-zero parallel 2-form. Also we obtain some equivalent conditions on a 3-dimensional normal almost contact metric manifold and we prove that if a 3-dimensional normal almost contact metric manifold which is not a ${\beta}$-Sasakian manifold satisfies cyclic parallel Ricci tensor, then the manifold is a manifold of constant curvature. Finally we prove the existence of such a manifold by a concrete example.

Generalized Quasi-Einstein Metrics and Contact Geometry

  • Biswas, Gour Gopal;De, Uday Chand;Yildiz, Ahmet
    • Kyungpook Mathematical Journal
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    • v.62 no.3
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    • pp.485-495
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    • 2022
  • The aim of this paper is to characterize K-contact and Sasakian manifolds whose metrics are generalized quasi-Einstein metric. It is proven that if the metric of a K-contact manifold is generalized quasi-Einstein metric, then the manifold is of constant scalar curvature and in the case of a Sasakian manifold the metric becomes Einstein under certain restriction on the potential function. Several corollaries have been provided. Finally, we consider Sasakian 3-manifold whose metric is generalized quasi-Einstein metric.

ON WEAKLY EINSTEIN ALMOST CONTACT MANIFOLDS

  • Chen, Xiaomin
    • Journal of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.707-719
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    • 2020
  • In this article we study almost contact manifolds admitting weakly Einstein metrics. We first prove that if a (2n + 1)-dimensional Sasakian manifold admits a weakly Einstein metric, then its scalar curvature s satisfies -6 ⩽ s ⩽ 6 for n = 1 and -2n(2n + 1) ${\frac{4n^2-4n+3}{4n^2-4n-1}}$ ⩽ s ⩽ 2n(2n + 1) for n ⩾ 2. Secondly, for a (2n + 1)-dimensional weakly Einstein contact metric (κ, μ)-manifold with κ < 1, we prove that it is flat or is locally isomorphic to the Lie group SU(2), SL(2), or E(1, 1) for n = 1 and that for n ⩾ 2 there are no weakly Einstein metrics on contact metric (κ, μ)-manifolds with 0 < κ < 1. For κ < 0, we get a classification of weakly Einstein contact metric (κ, μ)-manifolds. Finally, it is proved that a weakly Einstein almost cosymplectic (κ, μ)-manifold with κ < 0 is locally isomorphic to a solvable non-nilpotent Lie group.

PSEUDO-HERMITIAN MAGNETIC CURVES IN NORMAL ALMOST CONTACT METRIC 3-MANIFOLDS

  • Lee, Ji-Eun
    • Communications of the Korean Mathematical Society
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    • v.35 no.4
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    • pp.1269-1281
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    • 2020
  • In this article, we show that a pseudo-Hermitian magnetic curve in a normal almost contact metric 3-manifold equipped with the canonical affine connection ${\hat{\nabla}}^t$ is a slant helix with pseudo-Hermitian curvature ${\hat{\kappa}}={\mid}q{\mid}\;sin\;{\theta}$ and pseudo-Hermitian torsion ${\hat{\tau}}=q\;cos\;{\theta}$. Moreover, we prove that every pseudo-Hermitian magnetic curve in normal almost contact metric 3-manifolds except quasi-Sasakian 3-manifolds is a slant helix as a Riemannian geometric sense. On the other hand we will show that a pseudo-Hermitian magnetic curve γ in a quasi-Sasakian 3-manifold M is a slant curve with curvature κ = |(t - α) cos θ + q| sin θ and torsion τ = α + {(t - α) cos θ + q} cos θ. These curves are not helices, in general. Note that if the ambient space M is an α-Sasakian 3-manifold, then γ is a slant helix.

C12-SPACE FORMS

  • Gherici Beldjilali;Nour Oubbiche
    • Communications of the Korean Mathematical Society
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    • v.38 no.2
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    • pp.629-641
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    • 2023
  • The aim of this paper is two-fold. First, we study the Chinea-Gonzalez class C12 of almost contact metric manifolds and we discuss some fundamental properties. We show there is a one-to-one correspondence between C12 and Kählerian structures. Secondly, we give some basic results for Riemannian curvature tensor of C12-manifolds and then establish equivalent relations among 𝜑-sectional curvature. Concrete examples are given.