• 제목/요약/키워드: minimal submanifold

검색결과 33건 처리시간 0.019초

THE RIGIDITY OF MINIMAL SUBMANIFOLDS IN A LOCALLY SYMMETRIC SPACE

  • Cao, Shunjuan
    • 대한수학회보
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    • 제50권1호
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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 $S^4$.

ON THE STRUCTURE OF MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF NON-NEGATIVE CURVATURE

  • Yun, Gab-Jin;Kim, Dong-Ho
    • 대한수학회보
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    • 제46권6호
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    • pp.1213-1219
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    • 2009
  • Let M$^n$ be a complete oriented non-compact minimally immersed submanifold in a complete Riemannian manifold N$^{n+p}$ of nonnegative curvature. We prove that if M is super-stable, then there are no non-trivial L$^2$ harmonic one forms on M. This is a generalization of the main result in [8].

RIGIDITY OF MINIMAL SUBMANIFOLDS WITH FLAT NORMAL BUNDLE

  • Seo, Keom-Kyo
    • 대한수학회논문집
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    • 제23권3호
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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.

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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    • 제58권1호
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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.

SEMI-INVARIANT MINIMAL SUBMANIFOLDS OF CONDIMENSION 3 IN A COMPLEX SPACE FORM

  • Lee, Seong-Cheol;Han, Seung-Gook;Ki, U-Hang
    • 대한수학회논문집
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    • 제15권4호
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    • pp.649-668
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    • 2000
  • In this paper we prove the following : Let M be a real (2n-1)-dimensional compact minimal semi-invariant submanifold in a complex projective space P(sub)n+1C. If the scalar curvature $\geq$2(n-1)(2n+1), then m is a homogeneous type $A_1$ or $A_2$. Next suppose that the third fundamental form n satisfies dn = 2$\theta\omega$ for a certain scalar $\theta$$\neq$c/2 and $\theta$$\neq$c/4 (4n-1)/(2n-1), where $\omega$(X,Y) = g(X,øY) for any vectors X and Y on a semi-invariant submanifold of codimension 3 in a complex space form M(sub)n+1 (c). Then we prove that M has constant principal curvatures corresponding the shape operator in the direction of the distingusihed normal and the structure vector ξ is an eigenvector of A if and only if M is locally congruent to a homogeneous minimal real hypersurface of M(sub)n (c).

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GCR-LIGHTLIKE SUBMANIFOLDS OF A SEMI-RIEMANNIAN PRODUCT MANIFOLD

  • Kumar, Sangeet;Kumar, Rakesh;Nagaich, Rakesh Kumar
    • 대한수학회보
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    • 제51권3호
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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.

Screen Slant Lightlike Submanifolds of Indefinite Sasakian Manifolds

  • Haider, S.M. Khursheed;Advin, Advin;Thakur, Mamta
    • Kyungpook Mathematical Journal
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    • 제52권4호
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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 $R_2^9$.

ON SOME CR-SUBMANIFOLDS OF (n-1) CR-DIMENSION IN A COMPLEX PROJECTIVE SAPCE

  • Kwon, Jung-Hwan
    • 대한수학회논문집
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    • 제13권1호
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    • pp.85-94
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    • 1998
  • The purpose of this paper is to give sample characterizations of n-dimensional CR-submanifolds of (n-1) CR-semifolds of (n-1) CR-dimension immersed in a complex projective space $CP^{(n+p)/2}$ with Fubini-Study metric and we study an n-dimensional compact, orientable, minimal CR-submanifold of (n-1) CR-dimension in $CP^{(n+p)/2}$.

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