复射影空间中子流形的Gauss映照

本文利用复射影空间到欧氏空间的第一标准嵌入，对于复射影空间的子流形建立了一种广义的Ｇａｕｓｓ映照，并给出了这种广义的Ｇａｕ８ｓ映照是调和映照和相对仿射映照的条件。     

Submanifolds with Parallel Second Fundamental Form Studied via the Gauss Map

For an arbitrary n-dimensional Riemannian manifold N and an integer m ∈ {1,…,n−1} a covariant derivative on the Grassmann bundle ^ := G_m(T N) is introduced which has the property that an m-dimensional submanifold M ⊂ N has parallel second fundamental form if and only if its Gauss map M → ^ is affine. (For N Rⁿ this result was already obtained by J. Vilms in 1972.) By means of this relation a generalization of Cartan's theorem on the total geodesy of a geodesic umbrella can be derived: Suppose, initial data (p,W,b) prescribing a tangent space W ∈ G_m(T_pN) and a second fundamental form b at p ∈ N are given; for these data we construct an m-dimensional ‘umbrella’ M = M(p,W,b) ⊂ N the rays of which are helical arcs of N; moreover, we present tensorial conditions (not involving ) which guarantee that the umbrella M has parallel second fundamental form. These conditions are as well necessary, and locally every submanifold with parallel second fundamental form can be obtained in this way. Mathematics Subject Classifications (2000): 53B25, 53B20, 53B21.     

欧氏空间中具有共形Gauss映照的曲面

本文考虑欧氏空间中具有共形Ｇａｕｓｓ映照的曲面．从Ｇａｕｓｓ映照的观点给出了 ｖｅｒｏｎｅｓｅ曲面的一个新特征．     

Space-Like Surfaces in the Minkowski Space with Pointwise 1-Type Gauss Maps

Ukrainian Mathematical Journal - We first classify space-like surfaces in the Minkowski space , de Sitter space , and hyperbolic space ?3 with harmonic Gauss maps. Then we characterize and...     

On the Structure of Submanifolds with Degenerate Gauss Maps

An n-dimensional submanifold X of a projective space P ^N (C) is called tangentially degenerate if the rank of its Gauss mapping gamma;; X G(n, N) satisfies 0 < rank < n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space P ^N (C). By means of the focal images, three basic types of submanifolds are discovered: cones, tangentially degenerate hypersurfaces, and torsal submanifolds. Moreover, for tangentially degenerate submanifolds, a structural theorem is proven. By this theorem, tangentially degenerate submanifolds that do not belong to one of the basic types are foliated into submanifolds of basic types. In the proof the authors introduce irreducible, reducible, and completely reducible tangentially degenerate submanifolds. It is found that cones and tangentially degenerate hypersurfaces are irreducible, and torsal submanifolds are completely reducible while all other tangentially degenerate submanifolds not belonging to basic types are reducible.     

3维双曲空间中曲面的双曲Gauss映照和法Gauss映照

本文导出了3维双曲空间中曲面的双曲Gauss映照和法Gauss映照的关系,发现了一般的曲面由双曲Gauss映照和平均曲率函数唯一确定,并证明了双曲Gauss映照所满足的二阶线性椭圆方程,给出了两种形式的关于双曲Gauss映照的三阶非线性偏微分方程(组)的一个解.     

在Sn1+1空间中具有常数量曲率的类空超曲面的高斯映射

在这篇文章中,我们研究在de Sitter空间中具有非负常值的第r个平均曲率的紧致的类空超曲面.我们证明了在合适的条件下紧致的类空超曲面是全脐的.     

Submanifolds with totally umbilical Gauss image

The submanifolds whose Gauss images are totally umbilical submanifolds of the Grassmann manifold are under consideration. The main result is the following classification theorem: if the Gauss image of a submanifold F in a Euclidean space is totally umbilical then either the Gauss image is totally geodesic, or F is the surface in E ⁴ of the special structure. Submanifolds in a Euclidean space with totally geodesic Gauss image were classified earlier.     

On Submanifolds with Negative Curvature in Euclidean Space

In this paper we give a survey of results of different authors, among them Tenenblat, on the behaviour of submanifolds with negative curvature.     

关于Minkowski空间的子流形

利用Ｆｉｎｓｌｅｒ法曲率Ａ、Ｌａｎｄｓｂｅｒｇ曲率Ｌｙ、法切曲率Ｆｙ、Ｂｅｒｗａｌｄ联络Ｄ以及第二基本形式Ⅱ_ｙ，研究Ｍｉｎｋｏｗｓｋｉ空间中的子流形、子流形的旗曲率与李齐曲率．     

伪黎曼空间型的2一调和类空子流形

用活动标架法给出常曲率的伪黎曼流形的类空子流形为2-调和的充要条件,研究平均曲率为零的一些条件.     

保持高斯映射的仿射形变

给定Ｒｉｅｍａｎｎ流形到欧氏空间的仿射浸入ｆ：　Ｍｎ→ＲＮ　　，我们建立存在另一个与ｆ有相同高斯映射的仿射浸入ｆ：Ｍｎ→ＲＮ　　的条件，进一步利用这个条件，解答了仿射浸入的高斯映射将其确定到何种程度的问题．     

On the Gauss Map of Minimal Graphs

We consider graphs of solutions to the minimal surface equation which are unbounded over subarcs of the domain boundary. An extensive study of such surfaces was made by Jenkins and Serrin. In this note, properties of the Gauss map are studied.     

关于de Sitter空间中伪脐类空子流形

本文研究de Sitter空间中的紧致伪脐类空子流形,得到了这类空子流形的一个积分不等式及其一些刚性定理．     

复空间形式中全实2-调和子流形

本文研究了复空间形式中的全实2-调和子流形,获得了关于此类子流形的一些性质和一个Simons型积分不等式,并研究了其Pinching现象.