• 제목/요약/키워드: Null hypersurface

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GEOMETRY OF ISOPARAMETRIC NULL HYPERSURFACES OF LORENTZIAN MANIFOLDS

  • Ssekajja, Samuel
    • 대한수학회지
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    • 제57권1호
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    • pp.195-213
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    • 2020
  • We define two types of null hypersurfaces as; isoparametric and quasi isoparametric null hypersurfaces of Lorentzian space forms, based on the two shape operators associated with a null hypersurface. We prove that; on any screen conformal isoparametric null hypersurface, the screen geodesics lie on circles in the ambient space. Furthermore, we prove that the screen distributions of isoparametric (or quasi isoparametric) null hypersurfaces with at most two principal curvatures are generally Riemannian products. Several examples are also given to illustrate the main concepts.

INVARIANT NULL RIGGED HYPERSURFACES OF INDEFINITE NEARLY α-SASAKIAN MANIFOLDS

  • Mohamed H. A. Hamed;Fortune Massamba
    • 대한수학회논문집
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    • 제39권2호
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    • pp.493-511
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    • 2024
  • We introduce invariant rigged null hypersurfaces of indefinite almost contact manifolds, by paying attention to those of indefinite nearly α-Sasakian manifolds. We prove that, under some conditions, there exist leaves of the integrable screen distribution of the ambient manifolds admitting nearly α-Sasakian structures.

CANAL HYPERSURFACES GENERATED BY NON-NULL CURVES IN LORENTZ-MINKOWSKI 4-SPACE

  • Mustafa Altin;Ahmet Kazan;Dae Won Yoon
    • 대한수학회보
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    • 제60권5호
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    • pp.1299-1320
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    • 2023
  • In the present paper, firstly we obtain the general expression of the canal hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres, pseudo hyperbolic hyperspheres and null hyper-cones whose centers lie on a non-null curve with non-null Frenet vector fields in E41 and give their some geometric invariants such as unit normal vector fields, Gaussian curvatures, mean curvatures and principal curvatures. Also, we give some results about their flatness and minimality conditions and Weingarten canal hypersurfaces. Also, we obtain these characterizations for tubular hypersurfaces in E41 by taking constant radius function and finally, we construct some examples and visualize them with the aid of Mathematica.

SOME INTEGRATIONS ON NULL HYPERSURFACES IN LORENTZIAN MANIFOLDS

  • Massamba, Fortune;Ssekajja, Samuel
    • 대한수학회보
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    • 제56권1호
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    • pp.229-243
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    • 2019
  • We use the so-called pseudoinversion of degenerate metrics technique on foliated compact null hypersurface, $M^{n+1}$, in Lorentzian manifold ${\overline{M}}^{n+2}$, to derive an integral formula involving the r-th order mean curvatures of its foliations, ${\mathcal{F}}^n$. We apply our formula to minimal foliations, showing that, under certain geometric conditions, they are isomorphic to n-dimensional spheres. We also use the formula to deduce expressions for total mean curvatures of such foliations.

Some Results on Null Hypersurfaces in (LCS)-manifolds

  • Ssekajja, Samuel
    • Kyungpook Mathematical Journal
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    • 제59권4호
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    • pp.783-795
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    • 2019
  • We prove that a Lorentzian concircular structure (LCS)-manifold does not admit any null hypersurface which is tangential or transversal to its characteristic vector field. Due to the above, we focus on its ascreen null hypersurfaces and show that such hypersurfaces admit a symmetric Ricci tensor. Furthermore, we prove that there are no totally geodesic ascreen null hypersurfaces of a conformally flat (LCS)-manifold.

ON H2-PROPER TIMELIKE HYPERSURFACES IN LORENTZ 4-SPACE FORMS

  • Firooz Pashaie
    • 대한수학회논문집
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    • 제39권3호
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    • pp.739-756
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    • 2024
  • The ordinary mean curvature vector field 𝗛 on a submanifold M of a space form is said to be proper if it satisfies equality Δ𝗛 = a𝗛 for a constant real number a. It is proven that every hypersurface of an Riemannian space form with proper mean curvature vector field has constant mean curvature. In this manuscript, we study the Lorentzian hypersurfaces with proper second mean curvature vector field of four dimensional Lorentzian space forms. We show that the scalar curvature of such a hypersurface has to be constant. In addition, as a classification result, we show that each Lorentzian hypersurface of a Lorentzian 4-space form with proper second mean curvature vector field is C-biharmonic, C-1-type or C-null-2-type. Also, we prove that every 𝗛2-proper Lorentzian hypersurface with constant ordinary mean curvature in a Lorentz 4-space form is 1-minimal.

2-TYPE HYPERSURFACES SATISFYING ⟨Δx, x - x0⟩ = const.

  • Jang, Changrim
    • East Asian mathematical journal
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    • 제34권5호
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    • pp.643-649
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    • 2018
  • Let M be a connected n-dimensional submanifold of a Euclidean space $E^{n+k}$ equipped with the induced metric and ${\Delta}$ its Laplacian. If the position vector x of M is decomposed as a sum of three vectors $x=x_1+x_2+x_0$ where two vectors $x_1$ and $x_2$ are non-constant eigenvectors of the Laplacian, i.e., ${\Delta}x_i={\lambda}_ix_i$, i = 1, 2 (${\lambda}_i{\in}R$) and $x_0$ is a constant vector, then, M is called a 2-type submanifold. In this paper we proved that a connected 2-type hypersurface M in $E^{n+1}$ whose postion vector x satisfies ${\langle}{\Delta}x,x-x_0{\rangle}=c$ for a constant c, where ${\langle}$, ${\rangle}$ is the usual inner product in $E^{n+1}$, is of null 2-type and has constant mean curvature and scalar curvature.

TRANSNORMAL SYSTEMS ON $R_{1}^{n+1}$

  • Kwang Sung Park;Koon Chan Kim;Young Soo Jo
    • 대한수학회논문집
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    • 제12권1호
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    • pp.109-112
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    • 1997
  • In this paper, we study on a classification of hypersurfaces given by tansnormal functions on $R^{n+1}_1$. If M is a level set of a transnormal function on $R^{n+1}_1$, then it is one of a hyperplane, a cylinder around k-plane, a pseudo-sphere and a pseudo-hyperbolic space.

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