• Title/Summary/Keyword: Clifford minimal torus

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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.

SCALAR CURVATURE OF CONTACT CR-SUBMANIFOLDS IN AN ODD-DIMENSIONAL UNIT SPHERE

  • Kim, Hyang-Sook;Pak, Jin-Suk
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.3
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    • pp.541-549
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    • 2010
  • In this paper we derive an integral formula on an (n + 1)-dimensional, compact, minimal contact CR-submanifold M of (n - 1) contact CR-dimension immersed in a unit (2m+1)-sphere $S^{2m+1}$. Using this integral formula, we give a sufficient condition concerning with the scalar curvature of M in order that such a submanifold M is to be a generalized Clifford torus.

SCALAR CURVATURE OF CONTACT THREE CR-SUBMANIFOLDS IN A UNIT (4m + 3)-SPHERE

  • Kim, Hyang-Sook;Pak, Jin-Suk
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.3
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    • pp.585-600
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    • 2011
  • In this paper we derive an integral formula on an (n + 3)-dimensional, compact, minimal contact three CR-submanifold M of (p-1) contact three CR-dimension immersed in a unit (4m+3)-sphere $S^{4m+3}$. Using this integral formula, we give a sufficient condition concerning the scalar curvature of M in order that such a submanifold M is to be a generalized Clifford torus.