一类浸入在半黎曼卷积空间的完备超曲面

We deal with complete hypersurfaces immersed in a semi-Riemannian warped product of the type eI×f M~n,where M~n is a connected n-dimensional oriented Riemannian manifold.When the fiber M~n is complete with sectional curvature-k≤K_M for some positive constant k,under appropriate restrictions on the norm of the gradient of the height function h,we proceed with our technique in order to guarantee that complete hypersurface immersed in a semi-Riemannian warped product is a slice.Our approach is based on the well known generalized maximum principle and another suitable maximum principle at the infinity due to Yau.     

Minimal Translation Hypersurfaces in Euclidean Space

The graph of a function f defined in some open set of the Euclidean space of dimension (p + q) is said to be a translation graph if f may be expressed as the sum of two independent functions ? and ψ defined in open sets of the Euclidean spaces of dimension p and q, respectively. We obtain a useful expression for the mean curvature of the graph of f in terms of the Laplacian, the gradient of ? and ψ as well as of the mean curvatures of their graphs. We study translation graphs having zero mean curvature, that is, minimal translation graphs, by imposing natural conditions on ? and ψ, like harmonicity, minimality and eikonality (constant norm of the gradient), giving several examples as well as characterization results.     

The Rigidity of Hypersurfaces in Euclidean Space

In the present paper, the rigidity of hypersurfaces in Euclideanspace is revisited. The Darboux equation is highlighted and two newproofs of the rigidity are given via energy method and maximalprinciple, respectively.     

Minimal Hypersurfaces in Hyperbolic Spaces

In this paper, we reprove a theorem of M. Anderson [Invent. Math., 69 （1982）, pp. 477-494] which established the existence of a minimal hypersurface in the hyperbolic space with prescribed asymptotic boundary with non-negative mean curvature in the non-parametric case. We use the mean curvature flow method.     

On Pairs of Hypersurfaces in Euclidean Space

Mathematical Notes -     

G-Semidifferentiability in Euclidean Spaces

To unify the increasing number of generalized derivatives in the recent literature, a general concept of differentiability, embracing several classical cases, has been proposed in Ref. 1. Proceeding in this vein, the present note is devoted to deepening the study of the properties of this general approach and to point out, as much as possible, the connections with the Dini–Hadamard derivatives. The results obtained in this sense lay the groundwork for a future comparison among several different optimality conditions for constrained extremum problems.     

Ricci Soliton Biharmonic Hypersurfaces in the Euclidean Space

Ukrainian Mathematical Journal - We investigate biharmonic Ricci soliton hypersurfaces (Mn, g, ??, ?) whose potential field ?? satisfies certain conditions. We...     

Isometric immersions of Riemannian Spaces in Euclidean Spaces

Questions of the theory of isometric immersions of Riemannian spaces in Euclidean spaces beginning with the very first results onthis topic and also results on immersions of pseudo-Riemannian spaces in pseudo-Euclidean spaces and applications of the theory of immersions in the general theory of relativity are considered.Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 15, pp. 173–211, 1977.     

关于伪欧氏空间

本文提出了一些新概念，ｐ—伪转置阵，ｐ—伪正交阵，ｐ—伪对称阵，从而能以这些矩阵为工具研究伪欧氏空间的性质以及空间中两个特殊线性变换，伪正交变换和伪对称变换．本文得到的主要结果是定理２，定理３，定理４及定理６．本文还指出了北大编《高等代数》第二版中的伪正交变换习题的一个错误．     

Minimal Homogeneous Submanifolds in Euclidean Spaces

We prove that minimal (extrinsically) homogeneous submanifolds of the Euclidean space are totally geodesic. As an application, we obtain that a complex homogeneous submanifold of C ^N must be totally geodesic.     

The Hausdorff-Ershov Hierarchy in Euclidean Spaces

The topological arithmetical hierarchy is the effective version of the Borel hierarchy. Its class Δ^ta₂ is just large enough to include several types of pointsets in Euclidean spaces ℝ^k which are fundamental in computable analysis. As a crossbreed of Hausdorff's difference hierarchy in the Borel class Δ^B₂ and Ershov's hierarchy in the class Δ⁰₂ of the arithmetical hierarchy, the Hausdorff-Ershov hierarchy introduced in this paper gives a powerful classification within Δ^ta₂. This is based on suitable characterizations of the sets from Δ^ta₂ which are obtained in a close analogy to those of the Δ^B₂ sets as well as those of the Δ⁰₂ sets. A helpful tool in dealing with resolvable sets is contributed by the technique of depth analysis. On this basis, the hierarchy properties, in particular the strict inclusions between classes of different levels, can be shown by direct constructions of witness sets. The Hausdorff-Ershov hierarchy runs properly over all constructive ordinals, in contrast to Ershov's hierarchy whose denotation-independent version collapses at level ω². Also, some new characterizations of concepts of decidability for pointsets in Euclidean spaces are presented.     

Spherical Maps Of Euclidean Spaces

We classify the maps f: Rp → Rn suchthat ¦fat -fb¦ = ¦fbt -fc¦ whenever ¦a?b¦= ¦b? c¦.     

非平坦的Randers空间中的极小旋转超曲面

本文的研究分为两部分.第一部分是在特殊的Randers空间中得到了等距浸入的HT-极小旋转超曲面,这些特殊的Rander空间是非Minkowski的,但是它们的旗曲率为零.第二部分刻画了Funk空间中的各向异性极小旋转超曲面.     

欧氏空间半平行超曲面的分类

本文用活动标架法证明了Ｊ．Ｄｅｐｒｅｚ的关于欧氏空间半平行超曲面的局部分类定理．     

On 2-Quasi-Umbilical Hypersurfaces in Conformally Flat Spaces

The main result of this paper states that any 2-quasi-umbilical hypersurface of a semi-Riemannian conformally flat space is a manifold with pseudosymmetric Weyl tensor.     

Hypersurfaces of the Euclidean sphere with nonnegative Ricci curvature

In this paper we prove that a compact oriented hypersurface of a Euclidean sphere with nonnegative Ricci curvature and infinite fundamental group is isometric to an H(r)-torus with constant mean curvature. Furthermore, we generalize, whithout any hypothesis about the mean curvature, a characterization of Clifford torus due to Hasanis and Vlachos. Received: 19 March 2002