• Title, Summary, Keyword: warped product

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Contact CR-Warped product Submanifolds in Cosymplectic Manifolds

  • Atceken, Mehmet
    • Kyungpook Mathematical Journal
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    • v.56 no.3
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    • pp.965-977
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    • 2016
  • The aim of this paper is to study the geometry of contact CR-warped product submanifolds in a cosymplectic manifold. We search several fundamental properties of contact CR-warped product submanifolds in a cosymplectic manifold. We also give necessary and sufficient conditions for a submanifold in a cosymplectic manifold to be contact CR-(warped) product submanifold. After then we establish a general inequality between the warping function and the second fundamental for a contact CR-warped product submanifold in a cosymplectic manifold and consider contact CR-warped product submanifold in a cosymplectic manifold which satisfy the equality case of the inequality and some new results are obtained.

EINSTEIN WARPED PRODUCT SPACES

  • KIM, DONG-SOO
    • Honam Mathematical Journal
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    • v.22 no.1
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    • pp.107-111
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    • 2000
  • We study Einstein warped product spaces. As a result, we prove the following: if M is an Einstein warped product space with base a compact 2-dimensional surface, then M is simply a Riemannian product space.

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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).

CR-WARPED PRODUCT SUBMANIFOLDS OF NEARLY KAEHLER MANIFOLDS

  • Al-Luhaibi, Nadia S.;Al-Solamy, Falleh R.;Khan, Viqar Azam
    • Journal of the Korean Mathematical Society
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    • v.46 no.5
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    • pp.979-995
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    • 2009
  • As warped product manifolds provide an excellent setting to model space time near black holes or bodies with large gravitational field, the study of these manifolds assumes significance in general. B. Y. Chen [4] initiated the study of CR-warped product submanifolds in a Kaehler manifold. He obtained a characterization for a CR-submanifold to be locally a CR-warped product and an estimate for the squared norm of the second fundamental form of CR-warped products in a complex space form (cf [6]). In the present paper, we have obtained a necessary and sufficient conditions in terms of the canonical structures P and F on a CR-submanifold of a nearly Kaehler manifold under which the submanifold reduces to a locally CR-warped product submanifold. Moreover, an estimate for the second fundamental form of the submanifold in a generalized complex space is obtained and thus extend the results of Chen to a more general setting.

CHARACTERIZATION OF WARPED PRODUCT SUBMANIFOLDS OF LORENTZIAN CONCIRCULAR STRUCTURE MANIFOLDS

  • Hui, Shyamal Kumar;Pal, Tanumoy;Piscoran, Laurian Ioan
    • Communications of the Korean Mathematical Society
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    • v.34 no.4
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    • pp.1303-1313
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    • 2019
  • Recently Hui et al. ([8,9]) studied contact CR-warped product submanifolds and also warped product pseudo-slant submanifolds of a $(LCS)_n$-manifold $\bar{M}$. The characterization for both these classes of warped product submanifolds have been studied here. It is also shown that there do not exists any proper warped product bi-slant submanifold of a $(LCS)_n$-manifold. Although the existence of a bi-slant submanifold of $(LCS)_n$-manifold is ensured by an example.

ON A CLASSIFICATION OF WARPED PRODUCT SPACES WITH GRADIENT RICCI SOLITONS

  • Lee, Sang Deok;Kim, Byung Hak;Choi, Jin Hyuk
    • Korean Journal of Mathematics
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    • v.24 no.4
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    • pp.627-636
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    • 2016
  • In this paper, we study Ricci solitons, gradient Ricci solitons in the warped product spaces and gradient Yamabe solitons in the Riemannian product spaces. We obtain the necessary and sufficient conditions for the Riemannian product spaces to be Ricci solitons. Moreover we classify the warped product space which admit gradient Ricci solitons under some conditions of the potential function.

SOME WARPED PRODUCT SUBMANIFOLDS OF A KENMOTSU MANIFOLD

  • Khan, Viqar Azam;Shuaib, Mohammad
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.863-881
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    • 2014
  • Many differential geometric properties of a submanifold of a Kaehler manifold are conceived via canonical structure tensors T and F on the submanifold. For instance, a CR-submanifold of a Kaehler manifold is a CR-product if and only if T is parallel on the submanifold (c.f. [2]). Warped product submanifolds are generalized version of CR-product submanifolds. Therefore, it is natural to see how the non-triviality of the covariant derivatives of T and F gives rise to warped product submanifolds. In the present article, we have worked out characterizations in terms of T and F under which a contact CR- submanifold of a Kenmotsu manifold reduces to a warped product submanifold.

NONCONSTANT WARPING FUNCTIONS ON EINSTEIN LORENTZIAN WARPED PRODUCT MANIFOLDS

  • Jung, Yoon-Tae;Choi, Eun-Hee;Lee, Soo-Young
    • Honam Mathematical Journal
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    • v.40 no.3
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    • pp.447-456
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    • 2018
  • In this paper, we consider nonconstant warping functions on Einstein Lorentzian warped product manifolds $M=B{\times}_{f^2}F$ with an 1-dimensional base B which has a negative definite metric. As the results, we discuss that on M the resulting Einstein Lorentzian warped product metric is a future (or past) geodesically complete one outside a compact set.