局部对称流形的具常平均曲率的完备超曲面

本文研究局部对称黎曼流形中具常平均曲率的完备超曲面,得到了这类超曲面全脐的一个结果,其推广了文[7]和[4]中的结论.     

Lorentz流形中的类空超曲面

1.引言在[1]中,Calabi证明了n+1(n≤4)维Minkowski空间中的完备极大类空超曲面是全测地的。在[2]中 , Cheng-Yau对所有的n证明了这一结论。在[3]中,对于某一类Lorentz流形,Nishikawa证明了类似的结果。并且在[2]中,Cheng-Yau还证明了当具有常数平均曲率的类空超曲面M是Minkowski空间的闭子集时,有     

黎曼流形的一类变分问题与积分公式

<正> [1]中计算了常曲率空间中超曲面的平均曲率的任意函数的变分.本文把[1]的结果推广到常曲率空间的子流形和任意黎曼流形的超曲面.[1]的方法不能处理这两种情况.然后我们利用变分公式导出欧氏空间子流形的一组很一般的积分公式,包括 Minkowski公式,Gardner's 公式和[2]中诸公式作为特例.本文还讨论了变分问题(?)     

关于复射影空间的实极小超曲面

<正> 一、引言对于复射影空间的紧致实极小超曲面,Lawson H.B.和 Kon M.先后给出了第二基本形式长度平方(相当于超曲面的数量曲率)和截面曲率的 Pinching 定理.OkumuraM.又把[3]的结果推广到常平均曲率的实超曲面.本文的目的是进一步讨论这种量子化现象.     

纯量曲率和平均曲率成线性关系的完备超曲面

本文首先讨论纯量曲率R和平均曲率H满足R=kH,k=Const。的完备曲面帮超曲面。证得:R~3中这样的曲面是球面、柱面或平面;R~(n 1)中这样的超曲面当它是凸的且R有上界时是超球面、超平面或广义圆柱面;R~(2 p)(p≥1)中这样的曲面在一定附加条件下是包括球面和平面在内的四种曲面,最后讨论类似的子流形。这些结果改进或推广了[5,6,7]的结果。     

R^(5)中具有平行Fubini-Pick形式的Calabi超曲面的分类北大核心CSCD

Fubini-Pick形式关于Blaschke-Berwald度量(或中心仿射度量)平行的局部强凸等仿射(或中心仿射)超曲面分类问题,在过去十年内已被完全解决.文献[20]对Fubini-Pick形式关于Calabi度量平行的2,3维Calabi超曲面进行了分类,该文将其推广到4维的情形.     

具孤立奇点的映照的第二类相对上同调

<正> 在[6]中我们定义了两种相对上同调,并对在奇点处有有限重数的超曲面作了计算.G.-M.Greuel提出下述问题:[6]中哪些结果能推广到具孤立奇点的完全交叉.在[7]中我们把[6]中第一类相对上同调的计算推广到映照芽.本文把[6]中第二类相对上同调的计算推广到具孤立奇点的映照.     

关于具有常数中曲率曲面的一个定理

文献[1]证明了:若M~2是一个常曲率流形N中具有平行中曲率向量的曲面,那么M~2或者是N的一个全脐超曲面中的极小曲面,或者含于N中一个具有常数中曲率的3维脐子流形中. 把上述命题条件“具有平行中曲率向量”削弱为“具有常数中曲率”,[2]证明了下述     

关于拟常曲率流形的子流形的Simons型公式

对于浸入在常曲率流形中的超曲面,K.Nomizu and B.Smyth在[1]中计算其第二基本张量的长度平方的拉氏算子得到一个Simons型公式,运用这个公式,他们研究了在一些附加条件下R~(n 1)或S~(n 1)中超曲面的测定。J.Erbacher[2]和K.Yano and S.Ishihara[3]把[1]的结果推广到浸入在常曲率流形中余维为p(≥1)的子流形上去。本文把[2,3]的Simons型公式推广到拟常曲率流形的情形,用此公式我们求得拟常曲率流形的极小子流形为全测地子流形的一个充分条件,还给出这个公式的其他一些应用。     

CPn中的弱等焦Hopf超曲面

本文研究了 CPn中的弱等焦Hopf超曲面的分类,弱等焦超曲面首先是由滕楚莲与Thorbergsson在[13]中引入的.设M是CPn中的弱等焦Hopf超曲面,则M的不同焦半径的个数是2,4或6;并且当g=2或4时,M在CPn中是等参的;当g=6时,n=8,(M-)=π-1(M)与S7中的等参超曲面不Lie等价,此处π:S7→CP3是Hopf纤维化.     

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.     

The global structure of locally convex hypersurfaces in Finsler—Hadamard manifolds

Locally convex compact immersed hypersurfaces in the Finsler—Hadamard space with bounded T-curvature are considered. Under certain conditions on normal curvatures, such hypersurfaces are proved to be convex, embedded, and homeomorphic to the sphere. To this end, the Rauch theorem is generalized to exponential maps of hypersurfaces and the convexity of parallel hypersurfaces is proved.     

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.     

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.     

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.     

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.     

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.     

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.     

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.