• Title, Summary, Keyword: hypersurface

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ON A TOTALLY UMBILIC HYPERSURFACE OF FIRST ORDER

  • Kim, Jaeman
    • Honam Mathematical Journal
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    • v.39 no.4
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    • pp.465-473
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    • 2017
  • In this paper, we define a totally umbilic hypersurface of first order and show that a totally umbilic hypersurface of first order in an Einstein manifold has a parallel second fundamental form. Furthermore we prove that a complete, simply connected and totally umbilic hypersurface of first order in a space of constant curvature is a Riemannian product of Einstein manifolds. Finally we show a proper example which is a totally umbilic hypersurface of first order but not a totally umbilic hypersurface.

ON A HYPERSURFACE OF THE FIRST APPROXIMATE MATSUMOTO SPACE

  • Lee, Il-Yong;Jun, Dong-Gum
    • East Asian mathematical journal
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    • v.17 no.2
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    • pp.325-337
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    • 2001
  • We consider the special hypersurface of the first approximate Matsumoto metric with $b_i(x)={\partial}_ib$ being the gradient of a scalar function b(x). In this paper, we consider the hypersurface of the first approximate Matsumoto space with the same equation b(x)=constant. We are devoted to finding the condition for this hypersurface to be a hyperplane of the first or second kind. We show that this hypersurface is not a hyper-plane of third kind.

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ξ-PARALLEL STRUCTURE JACOBI OPERATORS OF REAL HYPERSURFACES IN A NONFLAT COMPLEX SPACE FORM

  • KIM, NAM-GIL;KI, U-HANG
    • Honam Mathematical Journal
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    • v.28 no.4
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    • pp.573-589
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    • 2006
  • Let M be a real hypersurface with almost contact metric structure $({\phi},{\xi},{\eta},g)$ in a non flat complex space form $M_n(c)$. In this paper, we prove that if the structure Jacobi operator $R_{\xi}$ is ${\xi}$-parallel and the Ricci tensor S commutes with the structure operator $\phi$, then a real hypersurface in $M_n(c)$ is a Hopf hypersurface. Further, we characterize such Hopf hypersurface in $M_n(c)$.

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SOME CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE (A) IN A NONFLAT COMPLEX SPACE FORM

  • Ki, U-Hang;Liu, Hui-Li
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.1
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    • pp.157-172
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    • 2007
  • In this paper, we prove that if the structure Jacobi operator $R_{\xi}-parallel\;and\;R_{\xi}$ commutes with the Ricci tensor S, then a real hypersurface with non-negative scalar curvature of a nonflat complex space form $M_{n}(C)$ is a Hopf hypersurface. Further, we characterize such Hopf hypersurface in $M_{n}(C)$.

REAL HYPERSURFACES WITH MIAO-TAM CRITICAL METRICS OF COMPLEX SPACE FORMS

  • Chen, Xiaomin
    • Journal of the Korean Mathematical Society
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    • v.55 no.3
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    • pp.735-747
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    • 2018
  • Let M be a real hypersurface of a complex space form with constant curvature c. In this paper, we study the hypersurface M admitting Miao-Tam critical metric, i.e., the induced metric g on M satisfies the equation: $-({\Delta}_g{\lambda})g+{\nabla}^2_g{\lambda}-{\lambda}Ric=g$, where ${\lambda}$ is a smooth function on M. At first, for the case where M is Hopf, c = 0 and $c{\neq}0$ are considered respectively. For the non-Hopf case, we prove that the ruled real hypersurfaces of non-flat complex space forms do not admit Miao-Tam critical metrics. Finally, it is proved that a compact hypersurface of a complex Euclidean space admitting Miao-Tam critical metric with ${\lambda}$ > 0 or ${\lambda}$ < 0 is a sphere and a compact hypersurface of a non-flat complex space form does not exist such a critical metric.

AN APPROACH FOR HYPERSURFACE FAMILY WITH COMMON GEODESIC CURVE IN THE 4D GALILEAN SPACE G4

  • Yoon, Dae Won;Yuzbasi, Zuhal Kucukarslan
    • The Pure and Applied Mathematics
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    • v.25 no.4
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    • pp.229-241
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    • 2018
  • In the present study, we derive the problem of constructing a hypersurface family from a given isogeodesic curve in the 4D Galilean space $G_4$. We obtain the hypersurface as a linear combination of the Frenet frame in $G_4$ and examine the necessary and sufficient conditions for the curve as a geodesic curve. Finally, some examples related to our method are given for the sake of clarity.

LINEAR WEINGARTEN HYPERSURFACES IN RIEMANNIAN SPACE FORMS

  • Chao, Xiaoli;Wang, Peijun
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.567-577
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    • 2014
  • In this note, we generalize the weak maximum principle in [4] to the case of complete linear Weingarten hypersurface in Riemannian space form $\mathbb{M}^{n+1}(c)$ (c = 1, 0,-1), and apply it to estimate the norm of the total umbilicity tensor. Furthermore, we will study the linear Weingarten hypersurface in $\mathbb{S}^{n+1}(1)$ with the aid of this weak maximum principle and extend the rigidity results in Li, Suh, Wei [13] and Shu [15] to the case of complete hypersurface.

A NOTE ON REAL HYPERSURFACES OF A COMPLEX SPACE FORM

  • Ki, U-Hang;Kim, He-Jin
    • Bulletin of the Korean Mathematical Society
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    • v.26 no.1
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    • pp.69-74
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    • 1989
  • Recently one of the present authors [2] asserted that a real hypersurface of a complex space form M$^{n}$ (c), c.neq.0, is of cyclic parallel if and only if AJ=JA and he showed also a complete and connected cyclic-parallel real hypersurface of M$^{n}$ (c), is congruent to type $A_{1}$, $A_{2}$ or A according as c>0 or c<0. A real hypersurface of a complex space form M$^{n}$ (c) is said to be covariantly cyclic constant if the cyclic sum of covariant derivative of the second fundamental form is constant. The purpose of the present paper is to extend theorem 3 and 4 in [2] when the hypersurfaces are of coveriantly cyclic constant, that is a real hypersurface of a complex space form M$^{n}$ (c), c.neq.0, is of covariantly cyclic constant if an only if AJ=JA, and a complete and connected covariantly cyclic constant real hypersurface of M$^{n}$ (c) is congruent to type $A_{1}$, $A_{2}$ or a according as c>0 or c<0.

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