关于超曲面上测度的Fourier变换的零点

设Ｓ是Ｅｕｃｌｉｄ空间Ｒ￣ｎ（ｎ≥２）中一个紧闭光滑超曲面，它关于原点中心对称，其Ｇａｕｓｓ曲率处处非零。设ｄμ是Ｓ上一个光滑正测度，是其Ｆｏｕｒｉｅｒ变换。本文证明，的零点集是一个紧集与可列多个微分同胚于单位球面的超曲面之无交并．     

关于超曲面上测度的Fourier交换的零点

设Ｓ是Ｅｕｃｌｉｄ空间Ｒ＾ｎ（ｎ≥２）中一个紧闭光滑超曲面，它关于原点中心对称，其Ｇａｕｓｓ曲率处处非零，设ｄμ是Ｓ上一个光滑正测度，ｄμ是其Ｆｏｕｒｉｅｒ变换，本文证明，ｄμ的零点集是一个紧集与可列多个微分同胚于单位球面的超曲面之无交并。     

欧几里得尺度作为明可夫斯基尺度的一个特征

如所知111,一个明可夫斯基尺度M.决定于尹,尸,…,尸的齐一次函数 F(才)二F(分‘,才2,…,分”).(1)更具体地观,在一个欧几里得空简E。里选取笛卡儿坐标系(.x‘),i~1,2,…,n,并考察超曲面 F(二)=1,(2)就是示标超曲面了.我们还假定了包舍坐标原点。为内点而且满足某些条件[a1,那末材。是以了为衡量标准的尺度空简. 特别是,在欧几里得尺度下的示标超曲面是以。为中心的单位超球;它的侧地核都是平面曲核,而且所在平面通过0. 我佣将舒渝逆简题:在明可夫斯基空简M,里,用它的尺度来锈导示标超曲面了上的一个尺度,从而产生了的侧地技;如果侧地核都…     

超曲面的共形数量不变量

对给定的共形流形及其中的超曲面,本文用Fefferman和Graham的辅助时空及其中齐次关联超曲面,引进了由齐次关联超曲面在辅助时空中的伪Riemann数量不变量诱导的原超曲面的共形数量不变量,提供了一套构建更多超曲面的共形数量不变量的计算方法,为寻找像Willmore方程一样关于超曲面的共形不变偏微分方程创造了路径.     

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.     

de Sitter空间中的类空超曲面在平均曲率下的形变

本文考虑了deSitter空间中紧致的类空扭曲面在平均曲率下的形变,证明了如果初始曲面满足某一Pinching条件,超曲面将收敛到一个球面．这一结论类似于球面上超曲面的情形,而与欧氏空间中的超曲面完全不同．     

Sn+1中具有调和M(o)bius曲率张量的超曲面

设x:Mn→Sn 1是(n 1)维单位球面Sn 1中的无脐点的超曲面.Sn 1中超曲面x有两个基本的共形不变量:M(o)bius度量g和M(o)bius第二基本形式B.当超曲面维数大于3时,在相差一个M(o)bius变换下这两个不变量完全决定了超曲面.另外M(o)bius形式Ф也是一个重要的不变量,在一些分类定理中Ф=0条件的假定是必要的.本文考虑了Sn 1(n≥3)中具有消失M(o)bius形式Ф的超曲面:对具有调和曲率张量的超曲面进行分类,进而,在M(o)bius度量的意义下,对Einstein超曲面和具有常截面曲率的超曲面也进行了分类.     

正定型超曲面上双曲闭特征的Maslov型指标的迭代公式

本文给出了R~(2n)中规范正定型超曲面上双曲闭特征的Maslov型指标的迭代公式.结果包含了凸超曲面和星形超曲面上已有的相应结论。     

THE CLOSED CURVES ON POSITIVE-TYPE HYPERSURFACES AND THE PERIODS OF NONLINEAR OSCILLATIONS

本文在Ｒｎ＋１中定义了一类超曲面并讨论了其上闭测地线长度的下界以及Ｈａｍｉｌｔｏｎ系统的周期解的最小周期，这类超曲面比星形超曲面广泛，因而本文结论包含了文献〔１〕、〔２〕、〔３〕中的相应结论     

完备λ-超曲面的几何

本文是一篇有关λ-超曲面的综述性文章,讨论了λ-超曲面的构造、稳定性和λ-超曲面的分类以及面积增长.     

On Holomorphic Automorphisms of a Class of Non-homogeneous Rigid Hypersurfaces in C^N＋1

The author determines the real-analytic infinitesimal CR automorphisms of a class of non-homogeneous rigid hypersurfaces in C^N＋1 near the origin, and the connected component containing the identity transformation of all locally holomorphic automorphisms of these hypersurfaces near the origin.     

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.     

Multiplicity of characteristics with Lagrangian boundary values on symmetric star-shaped hypersurfaces

In this paper, the multiplicity of Lagrangian orbits on C² smooth compact symmetric star-shaped hypersurfaces with respect to the origin in R2n is studied. These Lagrangian orbits begin from one Lagrangian subspace and end on another. An infinitely many existence result is proved via Z₂-index theory. This is a multiplicity result about the Arnold Chord Conjecture in some sense, and is a generalization of the problem about the multiplicity of Lagrangian orbits beginning from and ending on the same Lagrangian subspace which was considered in the authors' previous paper [F. Guo, C. Liu, Multiplicity of Lagrangian orbits on symmetric star-shaped hypersurfaces, Nonlinear Anal. 69 (4) (2008) 1425-1436].     

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.     

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.     

Lightlike Hypersurfaces of a Golden Semi-Riemannian Manifold

We introduce lightlike hypersurfaces of a golden semi-Riemannian manifold. We investigate several properties of lightlike hypersurfaces of a golden semi-Riemannian manifold. We prove that there is no radical anti-invariant lightlike hypersurface of a golden semi-Riemannian manifold. In particular, we obtain some results for screen semi-invariant lightlike hypersurfaces of a golden semi-Riemannian manifold. Moreover, we study screen conformal screen semi-invariant lightlike 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.     

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.     

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.     

Parallel and totally geodesic hypersurfaces of 5-dimensional 2-step homogeneous nilmanifolds

In this paper we study parallel and totally geodesic hypersurfaces of two-step homogeneous nilmanifolds of dimension five. We give the complete classification and explicitly describe parallel and totally geodesic hypersurfaces of these spaces. Moreover, we prove that two-step homogeneous nilmanifolds of dimension five which have one-dimensional centre never admit parallel hypersurfaces. Also we prove that the only two-step homogeneous nilmanifolds of dimension five which admit totally geodesic hypersurfaces have three-dimensional centre.