射影空间中代数曲线与超曲面的一些新的性质

发现了代数曲线的新的不变量一特征数,并得到了Pascal定理的不同于3次曲线的Cllasles定理和高次曲线中的Cayley-Bacharach定理等形式的高次推广.进一步研究了平面代数曲线的一些性质.通过定义m次Pascal超曲面,将Pascal定理推广到n维射影空间的m次超曲面中,证明了n-单纯形上的Pascal点位于一个m次Pascal超曲面的充要条件是其每个2维面上的Pascal点分别位于m次平面Pascal空间的一条代数曲线上.进一步,给出了一定条件下m次Pascal超曲面与m-1次Pascal超曲面之间的内在关系.     

双曲空间中的共形平坦级小超曲面

本文证明了双曲空间中共形平坦极小超曲面必为旋转超曲面或由一些旋转超曲面片用全测地超曲面片粘合而成，将这结果与王新民及许志才的一个已发表定理相组合，推广了Ｂｌａｉｒ关于推广悬链面的一个定理。     

双曲空间中的共形平坦极小超曲面

本文证明了双曲空间中的共形平坦极小超曲面必为旋转超曲面或由一些旋转超曲面片用全测地超曲面片粘合而成．将这结果与王新民及许志才的一个已发表定理相组合，推广了Ｂｌａｉｒ关于推广悬链面的一个定理．     

空间形式中具有两个线性相关平均曲率函数的超曲面

在空间形式中, 我们构造了一类泛函, 其临界点包括极小与r 极小超曲面. 给出了临界超曲面的代数、微分和变分刻画. 我们证明了Simons 类不存在定理: 在单位球面中不存在稳定的临界超曲面. 同时证明了Alexandrov 类存在性定理: 在欧氏空间中球面是唯一的稳定的临界超曲面.     

Anti de Sitter空间中Lorentzian超曲面的φ-伪球高斯映射

利用Legendrian对偶定理,证明了Anti de Sitter空间中的Lorentzian超曲面存在φ-伪球高斯映射,从而初步建立了Anti de Sitter空间中Lorentzian超曲面的斜几何.进一步的,证明了斜几何的基本定理,完成了φ~±-全脐超曲面的分类并给出了Lorentzian超曲面的φ~±-Anti de Sitter Weingarten型公式.     

Lorentz空间型中具有常数量曲率类空超曲面

研究了Lorentz空间型中具有常数量曲率的紧致类空超曲面,得到了这类超曲面的内蕴刚性定理.     

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曲面的的斜几何.     

空间形式中子流形的象半径与体积增长

本文讨论空间形式中子流形的象半径及体积增长的估计.作为应用,用r-平均曲率的语言获得了一个紧致超曲面的不存在性定理,它是经典紧致极小超曲面不存在性定理的推广.     

空间形式中子流形的象半径与体积增长

本文讨论空间形式中子流形的象半径及体积增长的估计。作为应用,用γ-平均曲率的语言获得了一个紧致超曲面的不存在性定理,它是经典紧致极小超曲面不存在性定理的推广.     

局部对称空间中具有常数量曲率的超曲面

利用自伴算子研究局部对称空间中具有常数量曲率的紧致超曲面,得到了这类超曲面中的某些刚性定理,推广了已有的结果.     

The structure vector field and structure Jacobi operator of real hypersurfaces in nonflat complex space forms

In this paper we determine the real hypersurfaces for which the structure Jacobi operator commutes over both the Ricci tensors and structure tensors (for a definition of the operator see Sect. 1). We prove that such hypersurfaces are homogneous real hypersurfaces of type (A) and are a special class of Hopf hypersurfaces.     

Non-embeddable real algebraic hypersurfaces

We study various classes of real hypersurfaces that are not embeddable into more special hypersurfaces in higher dimension, such as spheres, real algebraic compact strongly pseudoconvex hypersurfaces or compact pseudoconvex hypersurfaces of finite type. We conclude by stating some open problems.     

Circular trajectories on real hypersurfaces in a nonflat complex space form

In this paper we study which trajectories for Sasakian magnetic fields are circles on certain standard real hypersurfaces which are called hypersurfaces of type A in a nonflat complex space form. We also give a characterization of these real hypersurfaces by such a circular property of trajectories for Sasakian magnetic fields.     

Radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric

In this paper we introduce radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric. Such class of lightlike hypersurfaces cannot exist for indefinite almost Hermitian manifolds. The considered lightlike hypersurfaces have two important properties. The first one is the uniqueness of their screen distributions, which implies that the induced geometric objects are well-defined. The second property is that the induced Ricci tensor on radical transversal lightlike hypersurface of a Kähler manifold with Norden metric is symmetric. This allows to define an induced scalar curvature of the hypersurface. We obtain new results about lightlike hypersurfaces concerning their relations with non-degenerate hypersurfaces of almost complex manifolds with Norden metric. Examples of the considered hypersurfaces are given.     

Symmetries of Null Geometry in Indefinite Kenmotsu Manifolds

Null hypersurfaces have metrics with vanishing determinants and this degeneracy of these metrics leads to several difficulties. In this paper, null hypersurfaces of indefinite Kenmotsu space forms, tangent to the structure vector field, are studied with specific attention to locally symmetric, semi-symmetric and Ricci semi-symmetric null hypersurfaces. We show that locally symmetric and semi-symmetric null hypersurfaces are totally geodesic and parallel. These also hold for Ricci semi-symmetric null hypersurfaces, under a certain condition. We prove that, in null Einstein hypersurfaces of an indefinite Kenmotsu space form, tangent to the structure vector field, the local symmetry, semisymmetry and Ricci semi-symmetry notions are equivalent. For totally contact umbilical null hypersurfaces, we show that there are η-“Weyl” connections adapted to the induced structure on the null hypersurface.     

On Ruled Real Hypersurfaces in Complex Space Forms

Ruled real hypersurfaces of complex space forms are investigated by using the fact that such hypersurfaces can be constructed by moving a 1-codimensional complex totally geodesic submanifold of the ambient space along a curve. Among other results, a classification of minimal ruled real hypersurfaces and an example of a homogeneous ruled real hypersurface are given.     

Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian

We introduce various families of irreducible homaloidal hypersurfaces in projective space Pr, for all r?3. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves a question that has naturally arisen from the consideration of the classes of homaloidal hypersurfaces known so far. The result relies on a fine analysis of hypersurfaces that are dual to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hypersurfaces based on a certain degeneration of a generic Hankel matrix. The latter family fit non-classical versions of de Jonquières transformations. As a natural counterpoint, we broaden up aspects of the theory of Gordan-Noether hypersurfaces with vanishing Hessian determinant, bringing over some more precision into the present knowledge.     

Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces

We prove that a four-dimensional generalized symmetric space does not admit any non-degenerate hypersurfaces with parallel second fundamental form, in particular non-degenerate totally geodesic hypersurfaces, unless it is locally symmetric. However, spaces which are known as generalized symmetric spaces of type C do admit non-degenerate parallel hypersurfaces and we verify that they are indeed symmetric. We also give a complete and explicit classification of all non-degenerate totally geodesic hypersurfaces of spaces of this type.     

Einstein hypersurfaces of the Cayley projective plane

We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a certain radius; this completes the classification of Einstein hypersurfaces in rank-one symmetric spaces.     

On spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

The purpose of this paper is to study compact or complete spacelike hypersurfaces with constant normalized scalar curvature in a locally symmetric Lorentz space satisfying some curvature conditions. We give an optimal estimate of the squared norm of the second fundamental form of such hypersurfaces. Furthermore, the totally umbilical hypersurfaces are characterized.