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.     

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.     

Möbius hypersurfaces in S S+1with three distinct principal curvatures

On a hypersurface of a unit sphere without umbilical points, we know that three Möbius invariants can be defined and analogous to Euclidean case, we have the concepts of Möbius isoparametric and isotropic hypersurfaces. In this paper, we study the relationship between Euclidean geometry and Möbius geometry, and prove that a hypersurface in a sphere with constant length of the second fundamental form is Euclidean isoparametric if and only if it is Möbius isoparametric. When restricting to the case of three distinct principal curvatures, we show that such a hypersurface is either Möbius isoparametric or isotropic if the Blaschke tensor has constant eigenvalues.     

Real hypersurfaces in quaternionic projective space

Summary This paper is devoted to make a systematic study of real hypersurfaces of quaternionic projective space using focal set theory. We obtain three types of such real hypersurfaces. Two of them are known. Third type is new and in its study the first example of proper quaternion CR-submanifold appears. We study real hypersurfaces with constant principal curvatures and classify such hypersurfaces with at most two distinct principal curvatures. Finally we study the Ricci tensor of a real hypersurface of quaternionic projective space and classify pseudo-Einstein, almost-Einstein and Einstein real hypersurfaces.     

球面上具有四个不同主曲率的Moebius等参超曲面

设x:M→S~(n+1)(n≥5)是(n+1)-维单位球面上不含脐点的超曲面,在S~(n+1)的Moebius变换群下浸入x的四个基本不变量是:Moebius度量g;Moebius第二基本形式B;Moebius形式Φ和Blaschke张量A.本文给出S~(n+1)上具有重数1,1,1,m(m≥2)的四个不同Moebius主曲率的Moebius等参超曲面的分类.     

論保持平均曲率的黎曼測度V_m在歐氏空間E_(m+1)的變形

<正> §1.如所知,二維黎曼測度ds~2=E(u,ν)du~2+2F(u,ν)du dν+G(u,ν)dν~2;在E_3中的實現不是唯一的,而容有與單項目的兩個任意函數有關的連继變形.如此,曲面的測度不能決定曲面,但不同於此,m維黎曼測度(m≥3)如能在E_(m+1)中實現,一般地說實現它的超曲面只能是唯一的,也就是說,超曲面V_m E_(m+1)一般是不能變形的,而能變形的只是狹窄的一類超曲面.     

Problems of the anisotropy of strength

The concept of a strength tensor, analogous to the tensor of elastic constants, has been introduced with a view to using its components for approximate determination of laws governing the variation of strength of anisotropic materials in uniaxial tension or pure shear as a function of the orientation of the stress state. It has been shown that the plasticity condition postulated by Mises can be used as a criterion of strength of many "weakly" anisotropic materials, if one rejects the hypothesis according to which the condition is invariant in respect to hydrostatic pressure. In the case of "strongly" anisotropic materials, wood in particular, the Mises condition is at variance with the results obtained for variously oriented specimens tested in uniaxial tension. A strength condition in the form of a fourth-degree polynomial, based on the assumed existence of a strength tensor analogous to the tensor of elastic constants, has been postulated for these materials. The validity of this criterion has been demonstrated by constructing surfaces of equicritical plane stress states from experimental data obtained for pine wood, plywood, and glass-reinforced plastic laminates.Mekhanika Polimerov, Vol 1, No. 2, pp. 79–92, 1965     

Rigidity of complete hypersurfaces in the Euclidean space via anisotropic mean curvatures

In this paper, we study the rigidity of complete hypersurfaces in the Euclidean space Rⁿ⁺¹ via the so-called anisotropic mean curvatures, which correspond to a generalization of the usual mean curvatures. Under suitable constraints on the sign of the anisotropic mean curvatures and on the geometry of the height and angle functions naturally attached to such a hypersurface, we showthat it must be a hyperplane of Rⁿ⁺¹.     

R1^m+1中拟迷向类空超曲面

设f:M^m→R1^m+1是无脐点类空超曲面,则在Mm上可以定义四个基本的共形不变量:共形度量g,共形1-形式C,共形第二基本形式B,共形Blaschke张量A.如果存在光滑函数λ和常数μ,使得A+μB=Ag,则称M^m是拟迷向类空超曲面.本文不仅构造了拟迷向类空超曲面的例子,同时在相差R1^m+1的一个共形变换下,本文还完全分类了拟迷向类空超曲面.     

单位球面上仿Blaschke张量的特征值为常数的超曲面

设x:M~n→S~(n+1)是(n+1)-维单位球面上不含脐点的超曲面,在S~(n+1)的Moebius变换群下浸入x的四个基本不变量是:一个黎曼度量g称为Moebius度量;一个1-形式Φ称为Moebius形式;一个对称的(0,2)张量A称为Blaschke张量和一个对称的(0,2)张量B称为Moebius第二基本形式.对称的(0,2)张量D=A+λB也是Moebius不变量,称为浸入x的仿Blaschke张量,其中λ是常数,仿Blaschke张量的特征值称为仿Blaschke特征值.李海中和王长平(2003)研究了满足如下条件的超曲面:(i)Φ=0;(ii)存在可微函数λ和μ,使A+λg+μB=0.他们证明了λ和μ都是常数,并且给出了这类超曲面的分类,也就是D的特征值全相等的超曲面的分类.本文对满足如下条件的超曲面进行了分类:(i)Φ=0,(ii)对某一个常数λ,D具有两个互异的常数特征值.     

Gradient map of isoparametric polynomial and its application to Ginzburg-Landau system

In this note, we study properties of the gradient map of the isoparametric polynomial. For a given isoparametric hypersurface in sphere, we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications. We find that it should map not only the focal submanifolds to focal submanifolds, isoparametric hypersurfaces to isoparametric hypersurfaces, but also map isoparametric hypersurfaces to focal submanifolds. In particular, it turns out to be a homogeneous polynomial automorphism on certain isoparametric hypersurface. As an immediate consequence, we get the Brouwer degree of the gradient map which was firstly obtained by Peng and Tang with moving frame method. Following Farina's construction, another immediate consequence is a counterexample of the Brézis question about the symmetry for the Ginzburg-Landau system in dimension 6, which gives a partial answer toward the Open problem 2 raised by Farina.     

Infinitesimal CR automorphisms of rigid hypersurfaces in C2

In this paper, we describe the space of infinitesimal CR automorphisms of a rigid, real analytic, real hypersurface in C². We use these results to obtain a geometric characterization of the homogeneous hypersurfaces. Here, a hypersurface is called homogeneous if it is equivalent to one given by an equation of the formIm(w) =p wherep is a homogeneous polynomial inz and \(\bar z\) . This gives an answer in dimension 2 to a problem posed by Linda Rothschild. We give another answer, in terms of a normal form for the defining function, in our paper “A normal form for rigid hypersurfaces in C².”     

Lower bounds for eigenvalues of the Dirac-Witten operator

We get optimal lower bounds for the eigenvalues of the Dirac-Witten operator of compact(with or without boundary) spacelike hypersurfaces of Lorentian manifold satisfying certain conditions,just in terms of the mean curvature and the scalar curvature and the spinor energy-momentum tensor. In the limiting case,the spacelike hypersurface is either maximal and Einstein manifold with positive scalar curvature or Ricci-flat manifold with nonzero constant mean curvature.     

Classification of hypersurfaces with parallel M(o)bius second fundamental form in Sn+1

Let Mn(n≥2) be an immersed umbilic-free hypersurface in the(n+1)-dimensional unit sphere Sn+1. Then Mn is associated witha so-called M(o)bius metric g, and a M(o)bius second fundamental form Bwhich are invariants of Mn under the M(o)bius transformation groupof Sn+1.In this paper, we classify all umbilic-free hypersurfaces withparallel M(o)bius second fundamental form.     

An algebraic characterization of holomorphic nondegeneracy for real algebraic hypersurfaces and its application to CR mappings

We give a new algebraic characterization of holomorphic nondegeneracy for embedded real algebraic hypersurfaces in , . We then use this criterion to prove the following result about real analyticity of smooth CR mappings: any smooth CR mapping H between a real analytic hypersurface and a rigid polynomial holomorphically nondegenerate hypersurface is real analytic, provided the map H is not totally degenerate in the sense of Baouendi and Rothschild. Received September 19, 1997     

Equations of anisotropic elastodynamics as a symmetric hyperbolic system: Deriving the time-dependent fundamental solution

The dynamic system of anisotropic elasticity from three second order partial differential equations is written in the form of the time-dependent first order symmetric hyperbolic system with respect to displacement velocity and stress components. A new method of deriving the time-dependent fundamental solution of the obtained system is suggested in this paper. This method consists of the following. The Fourier transform image of the fundamental solution with respect to a space variable is presented as a power series expansion relative to the Fourier parameters. Then explicit formulae for the coefficients of these power series are derived successively. Using these formulae the computer calculation of fundamental solution components (displacement velocity and stress components arising from pulse point forces) has been made for general anisotropic media (orthorhombic and monoclinic) and the simulation of elastic waves has been obtained. These computational examples confirm the robustness of the suggested method.     

Biconservative surfaces in BCV‐spaces

Biconservative hypersurfaces are hypersurfaces with conservative stress‐energy tensor with respect to the bienergy functional, and form a geometrically interesting family which includes that of biharmonic hypersurfaces. In this paper we study biconservative surfaces in the 3‐dimensional Bianchi–Cartan–Vranceanu spaces, obtaining their characterization in the following cases: when they form a constant angle with the Hopf vector field; when they are SO(2)‐invariant.     

Ribaucour Transformations for Hypersurfaces in Space Forms

The theory of Ribaucour transformations for hypersurfaces in space forms is established. For any such hypersurface M, that admits orthonormal principal vector fields, it was shown the existence of a totally umbilic hypersurface locally associated to M by a Ribaucour transformation. A method of obtaining linear Weingarten surfaces in a three-dimensional space form is provided. By applying the theory, a new one-parameter family of complete constant mean curvature (cmc) surfaces in the unit sphere, locally associated to the flat torus, is obtained. The family contains a class of complete cmc cylinders in the sphere. In particular, one gets a family of complete minimal surfaces and minimal cylinders, locally associated to the Clifford torus.Mathematics Subject Classifications (2000): 53C20.