• Title/Summary/Keyword: Product manifolds

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NONCONSTANT WARPING FUNCTIONS ON EINSTEIN WARPED PRODUCT MANIFOLDS WITH 2-DIMENSIONAL BASE

  • Lee, Soo-Young
    • Korean Journal of Mathematics
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    • v.26 no.1
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    • pp.75-85
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    • 2018
  • In this paper, we study nonconstant warping functions on an Einstein warped product manifold $M=B{\times}_{f^2}F$ with a warped product metric $g=g_B+f(t)^2g_F$. And we consider a 2-dimensional base manifold B with a metric $g_B=dt^2+(f^{\prime}(t))^2du^2$. As a result, we prove the following: if M is an Einstein warped product manifold with a 2-dimensional base, then there exist generally nonconstant warping functions f(t).

ON A TOTALLY UMBILIC HYPERSURFACE OF FIRST ORDER

  • Kim, Jaeman
    • Honam Mathematical Journal
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    • v.39 no.4
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    • pp.465-473
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    • 2017
  • In this paper, we define a totally umbilic hypersurface of first order and show that a totally umbilic hypersurface of first order in an Einstein manifold has a parallel second fundamental form. Furthermore we prove that a complete, simply connected and totally umbilic hypersurface of first order in a space of constant curvature is a Riemannian product of Einstein manifolds. Finally we show a proper example which is a totally umbilic hypersurface of first order but not a totally umbilic hypersurface.

GEOMETRY OF HALF LIGHTLIKE SUBMANIFOLDS OF INDEFINITE KAEHLER MANIFOLDS WITH A QUARTER-SYMMETRIC METRIC CONNECTION

  • Gupta, Garima;Kumar, Rakesh
    • Communications of the Korean Mathematical Society
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    • v.35 no.3
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    • pp.979-998
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    • 2020
  • We study totally umbilical real half lightlike submanifolds of indefinite Kaehler manifolds with a quarter-symmetric metric connection. We obtain some conditions for a real half lightlike submanifold of an indefinite Kaehler manifold with a quarter-symmetric metric connection to be a product manifold. We derive the expression for induced Ricci type tensor 𝓡(0,2) and also obtain conditions for 𝓡(0,2) to be symmetric.

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 f-cosymplectic and (k, µ)-cosymplectic Manifolds Admitting Fischer -Marsden Conjecture

  • Sangeetha Mahadevappa;Halammanavar Gangadharappa Nagaraja
    • Kyungpook Mathematical Journal
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    • v.63 no.3
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    • pp.507-519
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    • 2023
  • The aim of this paper is to study the Fisher-Marsden conjucture in the frame work of f-cosymplectic and (k, µ)-cosymplectic manifolds. First we prove that a compact f-cosymplectic manifold satisfying the Fisher-Marsden equation R'*g = 0 is either Einstein manifold or locally product of Kahler manifold and an interval or unit circle S1. Further we obtain that in almost (k, µ)-cosymplectic manifold with k < 0, the Fisher-Marsden equation has a trivial solution.

SOME METRIC ON EINSTEIN LORENTZIAN WARPED PRODUCT MANIFOLDS

  • Lee, Soo-Young
    • Korean Journal of Mathematics
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    • v.27 no.4
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    • pp.1133-1147
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    • 2019
  • In this paper, let M = B×f2 F be an Einstein Lorentzian warped product manifold with 2-dimensional base. We study the geodesic completeness of some metric with constant curvature. First of all, we discuss the existence of nonconstant warping functions on M. As the results, we have some metric g admits nonconstant warping functions f. Finally, we consider the geodesic completeness on M.

SLANT SUBMANIFOLDS OF AN ALMOST PRODUCT RIEMANNIAN MANIFOLD

  • Sahin Bayram
    • Journal of the Korean Mathematical Society
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    • v.43 no.4
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    • pp.717-732
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    • 2006
  • In this paper, we study both slant 3nd semi-slant sub-manifolds of an almost product Riemannian manifold. We give characterization theorems for slant and semi-slant submanifolds and investigate special class of slant submanifolds which are product version of Kaehlerian slant submanifold. We also obtain integrability conditions for the distributions which are involved in the definition of a semi-slant submanifold. Finally, we prove a theorem on the geometry of leaves of distributions under a condition.

PRODUCT OF PL FIBRATORS AS CODIMENSION-k FIBRATORS

  • Im, Young-Ho;Kim, Yong-Kuk
    • Communications of the Korean Mathematical Society
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    • v.22 no.2
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    • pp.289-295
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    • 2007
  • We describe some conditions under which the product of two groups with certain property is a group with the same property, and we describe some conditions under which the product of hopfian manifolds is another hopfian manifold. As applications, we find some PL fibrators among the product of fibrators.

Non Existence of 𝒫ℛ-semi-slant Warped Product Submanifolds in a Para-Kähler Manifold

  • Sharma, Anil
    • Kyungpook Mathematical Journal
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    • v.60 no.1
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    • pp.197-210
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    • 2020
  • In this paper, we prove that there are no non-trivial 𝒫ℛ-semi-slant warped product submanifolds with proper slant coefficients in para-Kähler manifolds ${\bar{M}}$. We also present a numerical example that illustrates the existence of a 𝒫ℛ-warped product submanifold in ${\bar{M}}$.

WARPED PRODUCT SKEW SEMI-INVARIANT SUBMANIFOLDS OF LOCALLY GOLDEN RIEMANNIAN MANIFOLDS

  • Ahmad, Mobin;Qayyoom, Mohammad Aamir
    • Honam Mathematical Journal
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    • v.44 no.1
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    • pp.1-16
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    • 2022
  • In this paper, we define and study warped product skew semi-invariant submanifolds of a locally golden Riemannian manifold. We investigate a necessary and sufficient condition for a skew semi-invariant submanifold of a locally golden Riemannian manifold to be a locally warped product. An equality between warping function and the squared normed second fundamental form of such submanifolds is established. We also construct an example of warped product skew semi-invariant submanifolds.