• Title/Summary/Keyword: minimal submanifolds

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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.

Simons' Type Formula for Kaehlerian Slant Submanifolds in Complex Space Forms

  • Siddiqui, Aliya Naaz;Shahid, Mohammad Hasan;Jamali, Mohammed
    • Kyungpook Mathematical Journal
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    • v.58 no.1
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    • pp.149-165
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    • 2018
  • A. Bejancu [2] was the first who instigated the new concept in differential geometry, i.e., CR-submanifolds. On the other hand, CR-submanifolds were generalized by B. Y. Chen [7] as slant submanifolds. Further, he gave the notion of a Kaehlerian slant submanifold as a proper slant submanifold. This article has two objectives. For the first objective, we derive Simons' type formula for a minimal Kaehlerian slant submanifold in a complex space form. Then, by applying this formula, we give a complete classification of a minimal Kaehlerian slant submanifold in a complex space form and also obtain its some immediate consequences. The second objective is to prove some results about semi-parallel submanifolds.

GEOMETRIC CHARACTERISTICS OF GENERIC LIGHTLIKE SUBMANIFOLDS

  • Jha, Nand Kishor;Pruthi, Megha;Kumar, Sangeet;Kaur, Jatinder
    • Honam Mathematical Journal
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    • v.44 no.2
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    • pp.179-194
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    • 2022
  • In the present study, we investigate generic lightlike submanifolds of indefinite nearly Kaehler manifolds. After proving the existence of generic lightlike submanifolds in an indefinite generalized complex space form, a non-trivial example of this class of submanifolds is discussed. Then, we find a characterization theorem enabling the induced connection on a generic lightlike submanifold to be a metric connection. We also derive some conditions for the integrability of distributions defined on generic lightlike submanifolds. Further, we discuss the non-existence of mixed geodesic generic lightlike submanifolds in a generalized complex space form. Finally, we investigate totally umbilical generic lightlike submanifolds and minimal generic lightlike submanifolds of an indefinite nearly Kaehler manifold.

Screen Slant Lightlike Submanifolds of Indefinite Sasakian Manifolds

  • Haider, S.M. Khursheed;Advin, Advin;Thakur, Mamta
    • Kyungpook Mathematical Journal
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    • v.52 no.4
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    • pp.443-457
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    • 2012
  • In this paper, we introduce screen slant lightlike submanifold of an indefinite Sasakian manifold and give examples. We prove a characterization theorem for the existence of screen slant lightlike submanifolds. We also obtain integrability conditions of both screen and radical distributions, prove characterization theorems on the existence of minimal screen slant lightlike submanifolds and give an example of proper minimal screen slant lightlike submanifolds of $R_2^9$.

SLANT LIGHTLIKE SUBMANIFOLDS OF INDEFINITE NEARLY KAEHLER MANIFOLDS

  • Kumar, Tejinder;Kumar, Sangeet;Kumar, Pankaj
    • Honam Mathematical Journal
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    • v.43 no.2
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    • pp.239-258
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    • 2021
  • In the present paper, we introduce the study of slant lightlike submanifolds of indefinite nearly Kaehler manifolds. After proving some geometric results for the existence of slant lightlike submanifolds of indefinite nearly Kaehler manifolds, we give a non-trivial example of this class of lightlike submanifolds. Then, we derive some conditions for the integrability of the distributions associated with slant lightlike submanifolds of indefinite nearly Kaehler manifolds. Consequently, we study totally umbilical slant lightlike submanifolds of indefinite nearly Kaehler manifolds. Subsequently, we investigate minimal slant lightlike submanifolds of indefinite nearly Kaehler manifolds.

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.

SPHERICAL SUBMANIFOLDS WITH FINITE TYPE SPHERICAL GAUSS MAP

  • Chen, Bang-Yen;Lue, Huei-Shyong
    • Journal of the Korean Mathematical Society
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    • v.44 no.2
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    • pp.407-442
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    • 2007
  • The study of Euclidean submanifolds with finite type "classical" Gauss map was initiated by B.-Y. Chen and P. Piccinni in [11]. On the other hand, it was believed that for spherical sub manifolds the concept of spherical Gauss map is more relevant than the classical one (see [20]). Thus the purpose of this article is to initiate the study of spherical submanifolds with finite type spherical Gauss map. We obtain several fundamental results in this respect. In particular, spherical submanifolds with 1-type spherical Gauss map are classified. From which we conclude that all isoparametric hypersurfaces of $S^{n+1}$ have 1-type spherical Gauss map. Among others, we also prove that Veronese surface and equilateral minimal torus are the only minimal spherical surfaces with 2-type spherical Gauss map.

RIGIDITY OF MINIMAL SUBMANIFOLDS WITH FLAT NORMAL BUNDLE

  • Seo, Keom-Kyo
    • Communications of the Korean Mathematical Society
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    • v.23 no.3
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    • pp.421-426
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    • 2008
  • Let $M^n$ be a complete immersed super stable minimal submanifold in $\mathbb{R}^{n+p}$ with fiat normal bundle. We prove that if M has finite total $L^2$ norm of its second fundamental form, then M is an affine n-plane. We also prove that any complete immersed super stable minimal submanifold with flat normal bundle has only one end.

THE RIGIDITY OF MINIMAL SUBMANIFOLDS IN A LOCALLY SYMMETRIC SPACE

  • Cao, Shunjuan
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.1
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    • pp.135-142
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    • 2013
  • In the present paper, we discuss the rigidity phenomenon of closed minimal submanifolds in a locally symmetric Riemannian manifold with pinched sectional curvature. We show that if the sectional curvature of the submanifold is no less than an explicitly given constant, then either the submanifold is totally geodesic, or the ambient space is a sphere and the submanifold is isometric to a product of two spheres or the Veronese surface in $S^4$.