3维双曲空间中给定平均曲率曲面的Weierstrass表示

首先叙述双曲空间中曲面的Ｇａｕｓｓ映照的定义;导出Ｇａｕｓｓ映照所满足的Ｂｅｌｔｒａｍｉ方程;给出了给定平均曲率曲面的Ｗｅｉｅｒｓｔｒａｓｓ表示公式;并讨论了这种表示的完全可积条件．     

拟欧氏空间IR_n~(n 2)中类空曲面的Gaus映照

本文讨论拟欧氏空间ＩＲｎ＋２ｎ中的类空曲面，利用Ｇａｕｓ映照给出这类曲面的一般表示公式，从而推广了Ａｋｕｔａｇａｗａ和Ｎｉｓｈｉｋａｗａ的结果．     

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。     

G2,n中的可兼共形调和曲面

本文通过建立复Ｇｒａｓｓｍａｎｎ流形Ｇ２，ｎ中调和曲面的Ｇａｕｓｓ丛的基本公式，在适当的拓扑条件下给出了Ｇ２，ｎ中可兼共形调和曲面的构造定理．推广和改进了Ｂｕｒｓｔａｌｌ和Ｗｏｏｄ的低亏格曲面的结果．     

αK bH=c曲面的Bcklund变换

本文首先对Ｇａｕｓｓ曲率Ｋ和平均曲率Ｈ满足线性关系ａＫ＋ｂＨ＝ｃ的曲面，证明了其ＧａｕｓｓＷｅｉｎｇａｒｔｅｎ公式不仅可看作曲面的Ｇａｕｓｓ－Ｃｏｄａｚｚｉ方程的Ｌａｘ对，而且给出了Ｂ¨ａｃｋｌｕｎｄ变换．然后，将著名的关于负常曲率的Ｂ¨ａｃｋｌｕｎｄ定理推广到主曲率ｋ１和ｋ２满足关系（ｋ１－ｍ）（ｋ２－ｍ）＝－ｌ２的曲面     

调和Gauss映照和CP￣n中的1型子流形

本文证得ＣＰｎ中具有调和Ｇａｕｓ映照的子流形必是１型子流形，并给出一个彻底分类     

二项式系数与Gauss系数

二项式系数与Ｇａｕｓｓ系数万哲先（中国科学院）第二章Ｇａｕｓｓ系数２．１Ｇａｕｓｓ系数的定义和它的组合意义定义２．１设。是非负整数，Ｑ是４１的复数，而Ｘ是未定元，令我们把ｌ－ｆ－－ｉｆ叫做Ｇａｕｓｓ系数．注意，当ｍ一０时，Ｄ“Ｉ的分子和分母都是０个因...     

关于Gauss过程若干强极限定理

本文讨论了具有平稳增量的Ｇａｕｓｓ过程，在（０≤ｒ≤∞）时这类Ｇａｕｓｓ过程的增量有多小的问题．把有关的Ｗｉｅｎｅｒ过程的结果，在一定的条件下，推广到这类Ｇａｕｓｓ过程上去．此外还讨论了另一种形式的不可微模．     

作用在向量丛值Abel形式上的两个微分算子及其应用

本文定义了两个作用在向量丛值Ａｂｅｌ形式上的微分算子．运用这两个微分算子得到一些涉及ＣＰｎ中极小曲面的Ｇａｕｓｓ曲率与Ｋｈｌｅｒ角的Ｐｉｎｃｈｉｎｇ定理，并且讨论了邱成桐的第１０１问题．     

复Grassmann流形中伪全纯曲线的结构

设Ｍ是黎曼曲面，ｆ：Ｍ→Ｇ（ｍ，Ｎ）是全纯映照，我们证明了：若ｆ完全分裂，则由ｆ生成的伪全纯曲线是完全分裂的以及两维球面到复Ｇｒａｓｓｍａｎｎ流形的伪全纯曲线是完全分裂的．     

WEIERSTRASS REPRESENTATION FORSURFACES WITH PRESCRIBED NORMALGAUSS MAP AND GAUSS CURVATURE IN H~3

The author obtains a Weierstrass representation for surfaces with prescribed normal Gauss map and Gauss curvature in H3. A differential equation about the hyperbolic Gauss map is also obtained, which characterizes the relation among the hyperbolic Gauss map, the normal Gauss map and Gauss curvature. The author discusses the harmonicity of the normal Gauss map and the hyperbolic Gauss map from surface with constant Gauss curvature in H3 to S2 with certain altered conformal metric. Finally, the author considers the surface whose normal Gauss map is conformal and derives a completely nonlinear differential equation of second order which graph must satisfy.     

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.     

Duality with expanding maps and shrinking maps, and its applications to Gauss maps

We study expanding maps and shrinking maps of subvarieties of Grassmann varieties in arbitrary characteristic. The shrinking map was studied independently by Landsberg and Piontkowski in order to characterize Gauss images. To develop their method, we introduce the expanding map, which is a dual notion of the shrinking map and is a generalization of the Gauss map. Then we give a characterization of separable Gauss maps and their images, which yields results for the following topics: (1) Linearity of general fibers of separable Gauss maps; (2) Generalization of the characterization of Gauss images; (3) Duality on one-dimensional parameter spaces of linear subvarieties lying in developable varieties.     

The Gauss map of a hypersurface in Euclidean sphere and the spherical Legendrian duality

We investigate the Gauss map of a hypersurface in Euclidean n-sphere as an application of the theory of Legendrian singularities. We can interpret the image of the Gauss map as the wavefront set of a Legendrian immersion into a certain contact manifold. We interpret the geometric meaning of the singularities of the Gauss map from this point of view.     

Higher Gauss maps of Veronese varieties—A generalization of Boole’s formula and degree bounds for higher Gauss map images

The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gauss maps here was introduced by Fyodor L. Zak as a generalization of both ordinary Gauss maps and conormal maps. The main result is a closed formula for the degree of those images of Veronese varieties. This yields a generalization of a classical formula by George Boole on the degree of the dual varieties of Veronese varieties in 1844. As an application of our formula, degree bounds for higher Gauss map images of Veronese varieties are given.     

Willmore子流形的共形高斯映射

本文研究球空间中子流形的共形高斯映射,用Moebius不变量刻划了该映射 为调和映射的条件.作为特例,指出球空间的2维子流形的共形高斯映射是调和映射 当且仅当该子流形是Willmore子流形.     

The Gauss map of submanifolds in the Heisenberg group

We obtain criteria for the harmonicity of the Gauss map of submanifolds in the Heisenberg group and consider the examples demonstrating the connection between the harmonicity of this map and the properties of the mean curvature field. Also, we introduce a natural class of cylindrical submanifolds and prove that a constant mean curvature hypersurface with harmonic Gauss map is cylindrical.     

THE GAUSS MAP OF SUBMANIFOLDSIN SPACES OF CONSTANT CURVATURE

ＴＨＥＧＡＵＳＳＭＡＰＯＦＳＵＢＭＡＮＩＦＯＬＤＳＩＮＳＰＡＣＥＳＯＦＣＯＮＳＴＡＮＴＣＵＲＶＡＴＵＲＥ￥ＺＨＯＵＪＩＡＮＷＥＩＡｂｓｔｒａｃｔ：ＴｈｉｓｐａｐｅｒｓｔｕｄｉｅｓｔｈｅＧａｕｓｓｍａｐｏｆｓｕｂｍａｎｉｆｏｌｄｓｉｎｓｐａｃｅｆｏｒｍ...     

Vranceanu surface in $$ \mathbb{E}^4 $$ with pointwise 1- type Gauss map

In this article we investigate Vranceanu rotation surfaces with pointwise 1- type Gauss map in Euclidean 4-space $ \mathbb{E}^4 $ \mathbb{E}^4 . We show that a Vranceanu rotation surface M has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficent conditions for Vranceanu rotation surface to have pointwise 1-type Gauss map.