指标为2的半四维欧氏空间中三维Lorentzian子流形的Anti de Sitter高斯映射的奇点

作为R_(2~4)空间中子流形研究的补充工作,试图利用Lagrange奇点理论对R_(2~4)空间中三维Lorentzian子流形的微分几何性质及奇异性进行研究.     

5维伪欧氏空间中3维类时子流形的奇点分类

主要通过定义在指标数为2的5维伪欧氏空间中的3维类时子流形M上的类时高斯映射和类时高度函数,并研究M与管状超曲面CM的奇点分类.     

四维Minkowski空间中类空超曲面的双曲高斯映射的奇点

介绍了四维Minkowski空间中类空超曲面的局部理论,定义了类空超曲面上的双曲高斯映射,双曲高度函数及距离平方函数,给出了一些定理的详细证明.介绍了一种证明高度函数是Morse族的新方法并应用Arnold等建立的Lagrange奇点理论对类空超曲面的双曲高斯映射的奇点进行了分类.     

On the volume and the Gauss map image of spacelike hypersurfaces in de Sitter space

In this paper we prove that a complete spacelike hypersurface in de Sitter space such that its image under the Gauss map is contained in a hyperbolic geodesic ball of radius is necessarily compact and its -dimensional volume satisfies , where denotes the volume of a unitary round -sphere. We also characterize the case where these inequalities become equalities. As an application of our result, we also conclude that Goddard's conjecture is true under the assumption that the hyperbolic image of the hypersurface is bounded.

     

广义de Sitter空间中法向Lorentzian超曲面的nullcone Gauss映射的奇点

利用奇点理论研究了广义de Sitter空间中具有Lorentzian法空间的一类超曲面.介绍了这类超曲面的局部微分几何,定义了nullcone Gauss映射及nullcone高度函数族,进而研究了nullcone高度函数族的性质及nullcone高斯映射的几何意义,最后研究了这类超曲面的通有性质.     

n维Anti-de Sitter空间中类空超曲面的类时Gauss像的奇点

本文利用奇点理论研究了n维Anti-deSitter空间中的类空超曲面,介绍了类空超曲面的局部微分几何,定义了类时Anti-deSitterGauss像及Anti-deSitter高度函数,并进一步的利用Anti-deSitter高度函数族和流形间的切触理论研究了类时Anti-deSitterGauss像的几何意义及类空超曲面的通有性.最后研究了类空超曲面的AdS-Monge型.     

Anti de Sitter horospherical flat timelike surfaces

We investigate a special timelike surfaces in Anti de Sitter 3-space.We call such a timelike surface an Anti de Sitter horospherical flat surface which belongs to a class of surfaces given by one parameter families of Anti de Sitter horocycle.We give a generic classification of singularities and study the geometric properties of such surfaces from the viewpoint of Legendrian singularity theory.     

广义de Sitter空间中的类时超曲面

利用奇点理论研究广义de Sitter空间中的类时超曲面.介绍类时超曲面的局部微分几何,定义了广义de Sitter高斯像及广义de Sitter高度函数,研究广义deSitter高度函数族的性质及广义de Sitter高斯像的几何意义,介绍了一种证明高度函数为Morse族的新方法.最后研究了类时超曲面的通有性质.     

在Sn1+1空间中具有常数量曲率的类空超曲面的高斯映射

在这篇文章中,我们研究在de Sitter空间中具有非负常值的第r个平均曲率的紧致的类空超曲面.我们证明了在合适的条件下紧致的类空超曲面是全脐的.     

广义de Sitter空间中的类时超曲面的切触性质

从切触几何及Legendrian奇点理论的角度研究了广义de sitter空间中的类时超曲面的切触性质及gdS-高斯像的奇点的分类和几何意义.     

The singularities of null surfaces in Anti de Sitter 3-space

We study the geometric properties of degenerate surfaces, which are called AdS null surfaces, in Anti de Sitter 3-space from a contact viewpoint. These surfaces are associated to spacelike curves in Anti de Sitter 3-space. We define a map which is called the torus Gauss image. We also define two families of functions and use them to investigate the singularities of AdS null surfaces and torus Gauss images as applications of singularity theory of functions.     

Spacelike submanifolds of codimension two in anti-de Sitter space

In this paper, we study spacelike submanifolds of codimension two in anti-de Sitter space from the viewpoint of Legendrian singularity theory. We introduce the notion of the anti-de Sitter normalized Gauss map which is a generalization of the ordinary notion of Gauss map of hypersurfaces in Euclidean space. We also introduce the AdS-normalized Gauss–Kronecker curvature for a spacelike submanifold of codimention two in anti-de Sitter space. In the local sense, this curvature describes the contact of submanifolds with some model surfaces.     

de Sitter 空间中的常平均曲率完备类空超曲面

设Ｍ是常曲率ｃ的ｄｅ Ｓｉｔｔｅｒ空间Ｓ１＾ｎ＋１（ｃ）的常平均曲率的完备类空超曲面,Ｓ表示第二形式的范数平方。本文证明：差Ｓ〈２√ｎ－１ｃ,则Ｍ是全脐的和等距于一球面。     

Pseudo-hyperbolic Gauss maps of Lorentzian surfaces in anti-de Sitter space

In this paper, we determine the type numbers of the pseudo-hyperbolic Gauss maps of all oriented Lorentzian surfaces of constant mean and Gaussian curvatures and non-diagonalizable shape operator in the 3-dimensional anti-de Sitter space. Also, we investigate the behavior of type numbers of the pseudo-hyperbolic Gauss map along the parallel family of such oriented Lorentzian surfaces in the 3-dimensional anti-de Sitter space. Furthermore, we investigate the type number of the pseudo-hyperbolic Gauss map of one of Lorentzian hypersurfaces of B-scroll type in a general dimensional anti-de Sitter space.     

The Geometry of Lightlike Hypersurfaces of the de Sitter Space

It is proved that the geometry of lightlike hypersurfaces of the de Sitter space Sn+11 is directly connected with the geometry of hypersurfaces of the conformal space Cn. This connection is applied for a construction of an invariant normalization and an invariant affine connection of lightlike hypersurfaces as well as for studying singularities of lightlike hypersurfaces.     

Mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space

It is proved that, in Minkowski 3-space, a CSM-helicoidal surface, i.e., a helicoidal surface under cubic screw motion is isometric to a rotation surface so that helices on the helicoidal surface correspond to parallel circles on the rotation surface. By distinguishing a CSM-helicoidal surface as three cases, that is, the case of type I, the case of type II with negative and positive pitch, the relations are discussed between the mean curvatures or Gauss maps of a pair of isometric helicoidal and rotation surface. A CSM-helicoidal surface of Case 1 or 2 and its isometric rotation surface with null axis have same mean curvatures (resp. Gauss maps) if and only if they are minimal. But each pair of isometric CSM-helicoidal surface of Case 3 and rotation surface with spacelike axis have different Gauss maps.     

Remarks on Complete Spacelike Hyper-surfaces with Constant Mean Curvature in the de Sitter Space

Let M be an n(≥3)-dimensional completely non-compact spacelike hypersurface in the de Sitter space S1 (n+1) (1) with constant mean curvature and non negative sectional curvature. It is proved that M is isometric to a hyperbolic cylinder or an Euclidean space if H ≥1. When 2(n-1)~(1.2)/n < H < 1, there exists a complete rotation hypersurfaces which is not a hyperbolic cylinder.     

Singularities of lightlike hypersurface in semi-Euclidean 4-space with index 2

Anti de Sitter space is a maximally symmetric, vacuum solution of Einstein’s field equation with an attractive cosmological constant, and is the hyperquadric of semi-Euclidean space with index 2. So it is meaningful to study the submanifold in semi-Euclidean 4-space with index 2. However, the research on the submanifold in semi-Euclidean 4-space with index 2 has not been found from theory of singularity until now. In this paper, as a generalization of the study on lightlike hypersurface in Minkowski space and a preparation for the further study on anti de Sitter space, the singularities of lightlike hypersurface and Lorentzian surface in semi- Euclidean 4-space with index 2 will be studied. To do this, we reveal the relationships between the singularity of distance-squared function and that of lightlike hypersurface. In addition some geometric properties of lightlike hypersurface and Lorentzian surface are studied from geometrical point of view.     

Timelike surfaces of constant mean curvature ±1 in anti-de Sitter 3-space ℍ3 1(−1)

It is shown that timelike surfaces of constant mean curvature ± in anti-de Sitter 3-space ?³ ₁(?1) can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in ?SL₂? via Bryant type representation formulae. These Bryant type representation formulae are used to investigate an explicit one-to-one correspondence, the so-called Lawson–Guichard correspondence, between timelike surfaces of constant mean curvature ± 1 and timelike minimal surfaces in Minkowski 3-space E ³ ₁. The hyperbolic Gauß map of timelike surfaces in ?³ ₁(?1), which is a close analogue of the classical Gauß map is considered. It is discussed that the hyperbolic Gauß map plays an important role in the study of timelike surfaces of constant mean curvature ± 1 in ?³ ₁(?1). In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gauß map and timelike surface of constant mean curvature ± 1 in ?³ ₁(?1) is studied.