关于3维Minkowski空间中类空曲面的若干结果

在本文，我们证明２维黎曼流形能实现为Ｌ３中的类空极大曲面的充要条件是相应的Ｒｉｃｃｉ条件成立，此外还确定了Ｌ３中其平均曲率向量ｈ满足条件△ｈ＝λｈ（λ∈Ｒ）的类空曲面     

Extended Rectifying Curves in Minkowski 3-Space

In this paper, we consider nonhelical rectifying curves using the any orthonormal moving frame in Minkowski 3?space. Then, we give some relations between nonhelical rectifying curves and their Darboux vectors. Also, we demonstrate that the modified Darboux vectors of the curves are rectifying curves. So, this study is a general expression for the known rectifying curves. In the last section, we give some examples and draw their pictures.     

Symmetries of Real Hypersurfaces in Complex 3-Space

The main result of the paper consists in the proof of the fact that for any germ of a real analytic hypersurface in complex 3-space the following alternative (dimension conjecture) takes place: either the dimension of the group of holomorphic symmetries of the germ is at most the dimension of that of a nondegenerate hyperquadric (the latter equals 15), or the group is infinite-dimensional. We also discuss mistakes found in A. Ershova’s paper.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 171–179.Original Russian Text Copyright © 2005 by V. K. Beloshapka.     

Biharmonic Hypersurfaces in a Conformally Flat Space

Biharmonic hypersurfaces in a generic conformally flat space are studied in this paper. The equation of such hypersurfaces is derived and used to determine the conformally flat metric ${f^{-2}\delta_{ij}}$ on the Euclidean space ${\mathbb{R}^{m+1}}$ so that a minimal hypersurface ${M^{m}\longrightarrow (\mathbb{R}^{m+1}, \delta_{ij})}$ in a Euclidean space becomes a biharmonic hypersurface ${M^m\longrightarrow (\mathbb{R}^{m+1}, f^{-2}\delta_{ij})}$ in the conformally flat space. Our examples include all biharmonic hypersurfaces found in Ou (Pac J Math 248(1):217–232, 2010) and Ou and Tang (Mich Math J 61:531–542, 2012) as special cases.     

The Sub-Parabolic Lines in the Minkowski 3-Space

Results in Mathematics - In this paper, we investigate the focal surfaces obtained by the normal rectilinear congruence in the Minkowski 3-space. The sub-parabolic points and the ridge points are...     

Space-like Isoparametric Hypersurfaces in Lorentzian Space Forms

The purpose of this paper is to make clear the so-called Nomizu problem, whether it is possible to find examples of space-like isoparametric hypersurfaces in H ₁ ⁿ⁺¹ with more than two distinct principal curvatures. It is proved that a space-like isoparametric hypersurface in H ₁ ⁿ⁺¹ or S ₁ ⁿ⁺¹ can have at most two distinct principal curvatures. The authors present the classification and explicit analytic expressions of such type of isoparametric hypersurfaces. This paper was translated from J. Nanchang Univ. Nat. Sci. Ed., 2004, 28(2): 113–117     

Hypersurfaces of restricted type in Minkowski space

A submanifold M _n ^r of Minkowski space is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of to the tangent space of M _n ^r at every point of M _n ^r . In this paper we completely classify hypersurfaces of restricted type in . More precisely, we prove that a hypersurface of is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S ^k × , S _k ¹ × , H ^k × , S _n ¹ , H ⁿ , with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium.     

开零锥中洛伦兹超曲面的几何性质

开零锥中洛伦兹超曲面的研究难点在于无法利用伪外积运算从切空间中获取该超曲面的法向量.为了解决这一难题,可以借助Legendrian对偶定理的帮助.利用Legendrian对偶定理,构造了开零锥中的Lorentzian超曲面nullcone高斯像,Anti de Sitter高斯像和伪球高斯像并初步建立了开零锥中洛伦兹超...     

Ricci Soliton Biharmonic Hypersurfaces in the Euclidean Space

Ukrainian Mathematical Journal - We investigate biharmonic Ricci soliton hypersurfaces (Mn, g, ??, ?) whose potential field ?? satisfies certain conditions. We...     

Hypersurfaces of prescribed mean curvature in Lorentzian manifolds

We give a new existence proof for closed hypersurfaces of prescribed mean curvature in Lorentzian manifolds. Received April 12, 1999; in final form July 29, 1999 / Published online July 3, 2000     

Null Screen Isoparametric Hypersurfaces in Lorentzian Space Forms

In this paper, we develop the notion of screen isoparametric hypersurface for null hypersurfaces of Robertson–Walker spacetimes. Using this formalism we derive Cartan identities for the screen principal curvatures of null screen isoparametric hypersurfaces in Lorentzian space forms and provide a local characterization of such hypersurfaces.     

关于三维Minkowski空间中直线汇的一些注记

本文研究了三维Minkowski空间中直线汇的一些性质,特别是关于类时线汇的性质．讨论了线汇基本元素的存在性,并证明了关于三维Minkowski空间中类时线汇的配分参数的一个结果,推广了苏步青1927年的—个成果．     

三维Minkowski空间中的一类直线汇

研究了由三维Minkowski空间$E^3_1$中一个类空曲面$S_1$上一个单参数测地曲线族的切线所构成的直线汇$T$,它以$S_1$为一个焦曲面.证明了$T$的两个可展曲面族沿着第二个焦曲面$S_2$的正交曲线网相交的充要条件是$S_1$是可展曲面.对于$T$的两个焦曲面$S_1$和$S_2$之间沿着同一光线的对应,还证明了其保持渐近曲线网的充要条件.最后,研究了$T$的正交曲面$S$,并且证明了如果$S$是$E^3_1$中的一个极大曲面,那么,$T$的焦曲面$S_1$和$S_2$之间沿着同一光线的对     

Classifications of Isoparametric Hypersurfaces in Randers Space Forms

In this paper, we give the complete classifications of isoparametric hypersurfaces in Randers space forms. By studying the principal curvatures of anisotropic submanifolds in a Randers space (N, F) with the navigation data (h, W), we find that a Randers space form (N, F, dμ_BH) and the corresponding Riemannian space (N, h) have the same isoparametric hypersurfaces, but in general, their isoparametric functions are different. We give a necessary and sufficient condition for an isoparametric function of (N, h) to be isoparametric on (N, F, dμ_BH), from which we get some examples of isoparametric functions.     

Lorentz空间型中正常2-调和超曲面的分类

本文对Lorentz空间型中的正常2-调和超曲面进行了完全分类,它的形状算子的极小多项式的阶数至多是2.     

Curvature Estimates of Hypersurfaces in the Minkowski Space

A class of curvature estimates of spacelike admissible hypersurfaces related to translating solitons of the higher order mean curvature flow in the Minkowski space is obtained,which may ofer an idea to study an open question of the existence of hypersurfaces with the prescribed higher mean curvature in the Minkowski space.     

General Helices with Timelike Slope Axis in Minkowski 3-Space

In the present paper, we consider the general helices in Minkowski 3-space to have a constant timelike slope axis. As a result, we show that there exists no pseudo null helix with constant timelike slope axis or spacelike helix with timelike principal normal and constant timelike slope axis. Moreover, we obtain the parametric equations of spacelike helix with spacelike principal normal, timelike helix and null Cartan helix with timelike slope axis. We also give some related examples and their figures.     

Twisted Surfaces with Null Rotation Axis in Minkowski 3-Space

We examine curvature properties of twisted surfaces with null rotation axis in Minkowski 3-space. That is, we study surfaces that arise when a planar curve is subject to two synchronized rotations, possibly at different speeds, one in its supporting plane and one of this supporting plane about an axis in the plane. Moreover, at least one of the two rotation axes is a null axis. As is clear from its construction, a twisted surface generalizes the concept of a surface of revolution. We classify flat, constant Gaussian curvature, minimal and constant mean curvature twisted surfaces with a null rotation axis. Aside from pseudospheres, pseudohyperbolic spaces and cones, we encounter B-scrolls in these classifications. The appearance of B-scrolls in these classifications is of course the result of the rotation about a null axis. As for the cones in the classification of flat twisted surfaces, introducing proper coordinates, we prove that they are determined by so-called Clelia curves. With a Clelia curve we mean a curve that has linear dependent spherical coordinates.