• Title/Summary/Keyword: submanifolds

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Totally Umbilical Slant Lightlike Submanifolds of Indefinite Kaehler Manifolds

  • Sachdeva, Rashmi;Kumar, Rakesh;Bhatia, Satvinder Singh
    • Kyungpook Mathematical Journal
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    • v.57 no.3
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    • pp.503-516
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    • 2017
  • In this paper, we study totally umbilical slant lightlike submanifolds of indefinite Kaehler manifolds. We prove that there do not exist totally umbilical proper slant lightlike submanifolds in indefinite Kaehler manifolds other than totally geodesic proper slant lightlike submanifolds. We also prove that there do not exist totally umbilical proper slant lightlike submanifolds of indefinite Kaehler space forms. Finally, we give a characterization theorem on minimal slant lightlike submanifolds.

GEOMETRIC INEQUALITIES FOR SUBMANIFOLDS IN SASAKIAN SPACE FORMS

  • Presura, Ileana
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.4
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    • pp.1095-1103
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    • 2016
  • B. Y. Chen introduced a series of curvature invariants, known as Chen invariants, and proved sharp estimates for these intrinsic invariants in terms of the main extrinsic invariant, the squared mean curvature, for submanifolds in Riemannian space forms. Special classes of submanifolds in Sasakian manifolds play an important role in contact geometry. F. Defever, I. Mihai and L. Verstraelen [8] established Chen first inequality for C-totally real submanifolds in Sasakian space forms. Also, the differential geometry of slant submanifolds has shown an increasing development since B. Y. Chen defined slant submanifolds in complex manifolds as a generalization of both holomorphic and totally real submanifolds. The slant submanifolds of an almost contact metric manifolds were defined and studied by A. Lotta, J. L. Cabrerizo et al. A Chen first inequality for slant submanifolds in Sasakian space forms was established by A. Carriazo [4]. In this article, we improve this Chen first inequality for special contact slant submanifolds in Sasakian space forms.

GCR-LIGHTLIKE SUBMANIFOLDS OF INDEFINITE NEARLY KAEHLER MANIFOLDS

  • Kumar, Sangeet;Kumar, Rakesh;Nagaich, R.K.
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.4
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    • pp.1173-1192
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    • 2013
  • We introduce CR, SCR and GCR-lightlike submanifolds of indefinite nearly Kaehler manifolds and obtain their existence in indefinite nearly Kaehler manifolds of constant holomorphic sectional curvature $c$ and of constant type ${\alpha}$. We also prove characterization theorems on the existence of totally umbilical and minimal GCR-lightlike submanifolds of indefinite nearly Kaehler manifolds.

GCR-LIGHTLIKE SUBMANIFOLDS OF A SEMI-RIEMANNIAN PRODUCT MANIFOLD

  • Kumar, Sangeet;Kumar, Rakesh;Nagaich, Rakesh Kumar
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.883-899
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    • 2014
  • We introduce GCR-lightlike submanifold of a semi-Riemannian product manifold and give an example. We study geodesic GCR-lightlike submanifolds of a semi-Riemannian product manifold and obtain some necessary and sufficient conditions for a GCR-lightlike submanifold to be a GCR-lightlike product. Finally, we discuss minimal GCR-lightlike submanifolds of a semi-Riemannian product manifold.

f-BIHARMONIC SUBMANIFOLDS AND f-BIHARMONIC INTEGRAL SUBMANIFOLDS IN LOCALLY CONFORMAL ALMOST COSYMPLECTIC SPACE FORMS

  • Aslam, Mohd;Karaca, Fatma;Siddiqui, Aliya Naaz
    • Communications of the Korean Mathematical Society
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    • v.37 no.2
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    • pp.595-606
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    • 2022
  • In this paper, we have studied f-biharmonic submanifolds in locally conformal almost cosymplectic space forms and have derived condition on second fundamental form for f-biharmonic submanifolds. Also, we have discussed its integral submanifolds in locally conformal almost cosymplectic space forms.

ON GENERIC SUBMANIFOLDS OF LP-SASAKIAN MANIFOLDS WITH CONCURRENT VECTOR FIELDS

  • Ghosh, Sujoy;Jun, Jae-Bok;Sarkar, Avijit
    • Communications of the Korean Mathematical Society
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    • v.36 no.2
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    • pp.361-375
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    • 2021
  • The object of the present paper is to deduce some important results on generic submanifolds and also generic product of LP-Sasakian manifolds with concurrent vector fields. Also, we provide a necessary and sufficient condition for which the invariant distribution D and anti-invariant distribution D of M are Einstein. Also, we deduce an interesting necessary and sufficient condition for submanifolds of LP-Sasakian manifolds to be totally umbilical submanifolds. Especially we deal with the generic submanifolds admitting a Ricci soliton in LP-Sasakian manifolds endowed with concurrent vector fields.

A pointed blaschke manifold in euclidean space

  • Kim, Young-Ho
    • Journal of the Korean Mathematical Society
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    • v.31 no.3
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    • pp.393-400
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    • 1994
  • Subminifolds of Euclidean spaces have been studied by examining geodesics of the submanifolds viewed as curves of the ambient Euclidean spaces ([3], [7], [8], [9]). K.Sakamoto ([7]) studied submanifolds of Euclidean space whose geodesics are plane curves, which were called submanifolds with planar geodesics. And he completely calssified such submanifolds as either Blaschke manifolds or totally geodesic submanifolds. We now ask the following: If there is a point p of the given submanifold in Euclidean space such that every geodesic of the submanifold passing through p is a plane curve, how much can we say about the submanifold\ulcorner In the present paper, we study submanifolds of euclicean space with such property.

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