中心仿射曲面和余二维中心仿射浸入
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摘要
在1872年的“爱尔兰根纲领”中F.Klein把几何归结到可递变换群的几何不变量理论中,进而加以分类。这样,对每一个可递变换群,都可以定义一个隶属于这个可递变换群的几何。仿射空间Rn中的一个向量v经过变换之后变成向量v,如果它们之间的关系满足v=Av,A是一个n×n阶的非退化矩阵,则变换的全体就构成一个可递变换群,称为中心仿射变换群。仿射空间Rn中的中心仿射几何是隶属于Rn中的中心仿射变换群的几何,它研究的是图形在中心仿射变换下的不变性质。本文主要讨论了子流形在Rn+1(或Rn+2)中的中心仿射变换群G下的不变量和不变性质。
如果浸入x：M→Rn+1(或x：M→Rn+2)始终保持位置向量x截于切平面x*(TM),则在切丛TM上存在一个关于中心仿射变换群G不变的对称的2-形式g,g此时被称为x的中心仿射度量。对中心仿射浸入x：M→Rn+1,王长平计算了中心仿射超曲面的关于中心仿射度量g的第一和第二变分公式,得到一个新的中心仿射不变量-Tchebychev算子。刘会立和王长平进一步讨论了这个Tchebychev算子,得到了中心仿射Tchebychev曲面(Tchebychev超曲面)的分类。对中心仿射超曲面分类的工作一直是研究中心仿射浸入x：M→Rn+1的重要工作之一,而分类的方法主要是基于中心仿射不变量和这些不变量之间的关系。中心仿射浸入的环绕空间Rn+1中没有定义度量、中心仿射不变量之间的关系复杂以及中心仿射超曲面的范围广泛,这些都增加了中心仿射超曲面分类工作的难度。针对这种情况,本篇论文提出了一个解决方法,即先对所有的中心仿射超曲面分成一些大类,然后再考虑这些大类中的小类。本论文主要考虑了中心仿射超曲面中两大类——中心仿射平移曲面和中心仿射直纹曲面。在论文的第三章中,首先计算了中心仿射平移超曲面的基本结构方程和中心仿射不变量,根据这些公式和解相关的偏微分方程,得到了3维仿射空间中的数量曲率χ为常数的中心仿射平移曲面,Pick不变量J为常数的中心仿射平移曲面和||T||2为常数的中心仿射平移曲面,其中T是中心仿射Tchebychev向量场。对于中心仿射直纹超曲面,经过计算可以得到它的基本结构方程和一些不变量之间的关系。在3维仿射空间中,经过计算得到了中心仿射曲面的Pick不变量J、高斯曲率K、‖T‖2和中心仿射平均曲率H之间的关系,通过解相关的偏微分方程给出了3维仿射空间中的线性Weingarten中心仿射直纹曲面的详细分类。
对余二维中心仿射浸入x：Mn→Rn+2,一直存在两种不同的研究方式。一方面,Nomizu和Sasaki运用条件trh{T(X,Y)+h(SX,Y)}=0定义了一个预法化的Blaschke向量场(?)作为第二截向量场,其中h是中心仿射基本形式,S是Weingarten算子,T是
个2形式。另一方面,刘会立在研究余二维的中心仿射浸入时用一种新的方式定义了中心仿射度量g,然后选用△gx作为第二截向量场,其中△g表示度量g的拉普拉斯算子。这样在研究余二维的中心仿射浸入时,预法化的Blaschke向量场(?)和向量场△gx都可以作为第二截向量场,它们之间的联系和区别一直以来是大家所关注的问题。在论文的第四章中,首先运用活动标架法计算出余二维的中心仿射浸入x的基本公式和中心仿射不变量,根据这些公式和不变量,得到了这两个第二截向量场之间的关系即(?)=1/nΔgx-H/2x,其中H是中心仿射平均曲率,这样就统一了这两个第二截向量场,也就是统一了余二维中心仿射浸入的两种不同结构。接下来使用△gx作为第二截向量场,计算出了余二维的中心仿射浸入第一、第二变分公式,从而定义了余二维的极小中心仿射浸入。可以证明这样定义的余二维的极小中心仿射浸入也是预法化的Blaschke向量场(?)作为第二截向量场的极小中心仿射浸入。作为例子,在论文中验证了R4中Pick不变量恒为零的中心仿射齐性曲面是中心仿射极小的。
本文共分为四个部分：第一部分介绍了仿射微分几何和中心仿射几何的发展历程。第二部分列出了中心仿射微分几何的预备知识。第三部分是中心仿射超曲面,主要是关于R3中中心仿射平移曲面和中心仿射直纹面的一些结果。第四部分是关于余二维的中心仿射子流形,运用活动标架法计算了余二维的中心仿射子流形的第一和第二变分公式,并得到了不同的中心仿射法化之间的关系。
Felix Klein's1872Erlangen program puts geometry down to the geometric invariant theory of the transitive transformation group, and then categorizes them. Thus, there is a transitive transformation group, there is a geometry attached to this transitive transformation group. In affine space An, the vector v can be transformed to the vector v. If there exists a relation between v and v satisfying that v=Av, where A is an×n non-degenerate matrix, all transformations compose a group and this group is called as a centroaffine transformation group. Centroaffine geometry of affine space An is a branch of discussing invariance of graphics under centroaffine transformation, and it belongs to centroaffine transformation group of An. This paper considers the geometry attached to centroaffine transformation group, that is, geometric invariance of submanifolds under centroaffine transformation group G in Rn+1(or Rn+2).
If the immersions x:M→Rn+1(or x:M→Rn+2) always keeps position vector fields x transversal to tangent plane x*(TM), there exists a symmetric2-forms g with regard to centroaffine transformation group G, and g is called as the centroaffine metric. For centroaffine immersions x:M→Rn+1, Professor Wang Changping calculated the first and the second Variational formula of centroaffine hypersurfaces with respect to g, and obtained a new centroaffine invariant---Tchebychev operator. Professor Liu Huili and Professor Wang Changping further study of this Tchebychev operator, and got the classification of Tchebychev sufaces (Tchebychev hypersurfaces). From then on the classification of hypersurfacs became one of the important works of studing centroaffine immersions x:M→Rn+1, and the methods of the classification of hypersurfacs are mainly based on the centroaffine invariant and the relationship between them. Since there is not metric in ambient space Rn+1of centroaffine immersions, the calculation in ambient space Rn+1is very inconvenient. On the other hand, the complex relationship between centroaffine variants and wide range of centroaffine hypersurfaces make the work of the classification of centroaffine hypersurfacs difficult to carry on. In response to these situations, this paper poses a solution:firstly the all centroaffine hypersurfacs are divided into several broad categories, and then we get detail classifacation from the larger category. This paper considers two major categories of centroaffine hypersurfacs-centroaffine translaton surfaces and centroaffine ruled surfaces. In the third part of this paper, we get basic structure equations of centroaffine translation hypersurfaces and some centroaffine invariants. According to these equations and relational partial differential equation, we obtain centroaffine translation surfaces with constant Gauss curvature, centroaffine translation surfaces with constant Pick invariant, and centroaffine translation surfaces with||T||2=constant in R3. On the other hand, by solving certern partial differential equations we obtain some classification results for linear Weingarten centroaffine ruled surfaces in3-affine space R3and prove that a nondegenerate centroaffine ruled surface is centroaffine minimal if and only if its Gauss curvature is constant.
For centroaffine immersions of codimension two x:Mn→Rn+2, there always are two different research methods. Nomize and Sasaki defined a pre-normalized Blaschke vector field ξ using the conditiontrh{T(X,Y)+h(SX,Y)}=0, where h is the centroaffine basic form, S is the Weingarten operator, T is centroaffine2-form. On the other hand, Professor Liu has studied centroaffine immersions of codimension two using a new method to define the metric g, and chose Δgx as the second transversal vector fields, where Δg is the Laplacian of metric g. When we study centroaffine immersions of codimension two, ξ and Δgx all can be think as the second transversal vector field, and the relations and differences between them are the questions we must consider. In the fourth part of this paper, we calculate the basic equations and invariants centroaffine immersions of codimension two using moving frames. Depending on these equations and invariants we get the relation between these two second transversal vector fields, that is,ξ=1/nΔgx-H/2x, where H is centroaffine mean curvature. Then we compute the first and also second variational formulas of centroaffine volume integral using Δgx as the second transversal vector field and then define the minimal centroaffine immersions of codimension two, which is consistent to the minimal centroaffine immersions of codimension two using ξ as the second transversal vector field. As some examples, we proof that homogeneous surfaces with vanishing Pick invariant in R4are centroaffine minimal.
In this paper, we introduce the history of Differetial Geometry, especially about Affine Differetial Geometry (ADG) and Centroaffine Differetial Geometry in the first part. In the second part we describe some knowledge in Differetial Geometry, which is the basic of Centroaffine Differetial Geometry. Then in the third part we talk about centroaffine hypersurfaces. Mainly, we get some results about centroaffine translation surfaces and centroaffine ruled surfaces in R3. In the last part we introduce the centroaffine subminafolds of codimention two. Here we compute the first and the second formulas of area variantion and get the relation between different centroaffine normalizations.
如果浸入x：M→Rn+1(或x：M→Rn+2)始终保持位置向量x截于切平面x*(TM),则在切丛TM上存在一个关于中心仿射变换群G不变的对称的2-形式g,g此时被称为x的中心仿射度量。对中心仿射浸入x：M→Rn+1,王长平计算了中心仿射超曲面的关于中心仿射度量g的第一和第二变分公式,得到一个新的中心仿射不变量-Tchebychev算子。刘会立和王长平进一步讨论了这个Tchebychev算子,得到了中心仿射Tchebychev曲面(Tchebychev超曲面)的分类。对中心仿射超曲面分类的工作一直是研究中心仿射浸入x：M→Rn+1的重要工作之一,而分类的方法主要是基于中心仿射不变量和这些不变量之间的关系。中心仿射浸入的环绕空间Rn+1中没有定义度量、中心仿射不变量之间的关系复杂以及中心仿射超曲面的范围广泛,这些都增加了中心仿射超曲面分类工作的难度。针对这种情况,本篇论文提出了一个解决方法,即先对所有的中心仿射超曲面分成一些大类,然后再考虑这些大类中的小类。本论文主要考虑了中心仿射超曲面中两大类——中心仿射平移曲面和中心仿射直纹曲面。在论文的第三章中,首先计算了中心仿射平移超曲面的基本结构方程和中心仿射不变量,根据这些公式和解相关的偏微分方程,得到了3维仿射空间中的数量曲率χ为常数的中心仿射平移曲面,Pick不变量J为常数的中心仿射平移曲面和||T||2为常数的中心仿射平移曲面,其中T是中心仿射Tchebychev向量场。对于中心仿射直纹超曲面,经过计算可以得到它的基本结构方程和一些不变量之间的关系。在3维仿射空间中,经过计算得到了中心仿射曲面的Pick不变量J、高斯曲率K、‖T‖2和中心仿射平均曲率H之间的关系,通过解相关的偏微分方程给出了3维仿射空间中的线性Weingarten中心仿射直纹曲面的详细分类。
对余二维中心仿射浸入x：Mn→Rn+2,一直存在两种不同的研究方式。一方面,Nomizu和Sasaki运用条件trh{T(X,Y)+h(SX,Y)}=0定义了一个预法化的Blaschke向量场(?)作为第二截向量场,其中h是中心仿射基本形式,S是Weingarten算子,T是
个2形式。另一方面,刘会立在研究余二维的中心仿射浸入时用一种新的方式定义了中心仿射度量g,然后选用△gx作为第二截向量场,其中△g表示度量g的拉普拉斯算子。这样在研究余二维的中心仿射浸入时,预法化的Blaschke向量场(?)和向量场△gx都可以作为第二截向量场,它们之间的联系和区别一直以来是大家所关注的问题。在论文的第四章中,首先运用活动标架法计算出余二维的中心仿射浸入x的基本公式和中心仿射不变量,根据这些公式和不变量,得到了这两个第二截向量场之间的关系即(?)=1/nΔgx-H/2x,其中H是中心仿射平均曲率,这样就统一了这两个第二截向量场,也就是统一了余二维中心仿射浸入的两种不同结构。接下来使用△gx作为第二截向量场,计算出了余二维的中心仿射浸入第一、第二变分公式,从而定义了余二维的极小中心仿射浸入。可以证明这样定义的余二维的极小中心仿射浸入也是预法化的Blaschke向量场(?)作为第二截向量场的极小中心仿射浸入。作为例子,在论文中验证了R4中Pick不变量恒为零的中心仿射齐性曲面是中心仿射极小的。
本文共分为四个部分：第一部分介绍了仿射微分几何和中心仿射几何的发展历程。第二部分列出了中心仿射微分几何的预备知识。第三部分是中心仿射超曲面,主要是关于R3中中心仿射平移曲面和中心仿射直纹面的一些结果。第四部分是关于余二维的中心仿射子流形,运用活动标架法计算了余二维的中心仿射子流形的第一和第二变分公式,并得到了不同的中心仿射法化之间的关系。
Felix Klein's1872Erlangen program puts geometry down to the geometric invariant theory of the transitive transformation group, and then categorizes them. Thus, there is a transitive transformation group, there is a geometry attached to this transitive transformation group. In affine space An, the vector v can be transformed to the vector v. If there exists a relation between v and v satisfying that v=Av, where A is an×n non-degenerate matrix, all transformations compose a group and this group is called as a centroaffine transformation group. Centroaffine geometry of affine space An is a branch of discussing invariance of graphics under centroaffine transformation, and it belongs to centroaffine transformation group of An. This paper considers the geometry attached to centroaffine transformation group, that is, geometric invariance of submanifolds under centroaffine transformation group G in Rn+1(or Rn+2).
If the immersions x:M→Rn+1(or x:M→Rn+2) always keeps position vector fields x transversal to tangent plane x*(TM), there exists a symmetric2-forms g with regard to centroaffine transformation group G, and g is called as the centroaffine metric. For centroaffine immersions x:M→Rn+1, Professor Wang Changping calculated the first and the second Variational formula of centroaffine hypersurfaces with respect to g, and obtained a new centroaffine invariant---Tchebychev operator. Professor Liu Huili and Professor Wang Changping further study of this Tchebychev operator, and got the classification of Tchebychev sufaces (Tchebychev hypersurfaces). From then on the classification of hypersurfacs became one of the important works of studing centroaffine immersions x:M→Rn+1, and the methods of the classification of hypersurfacs are mainly based on the centroaffine invariant and the relationship between them. Since there is not metric in ambient space Rn+1of centroaffine immersions, the calculation in ambient space Rn+1is very inconvenient. On the other hand, the complex relationship between centroaffine variants and wide range of centroaffine hypersurfaces make the work of the classification of centroaffine hypersurfacs difficult to carry on. In response to these situations, this paper poses a solution:firstly the all centroaffine hypersurfacs are divided into several broad categories, and then we get detail classifacation from the larger category. This paper considers two major categories of centroaffine hypersurfacs-centroaffine translaton surfaces and centroaffine ruled surfaces. In the third part of this paper, we get basic structure equations of centroaffine translation hypersurfaces and some centroaffine invariants. According to these equations and relational partial differential equation, we obtain centroaffine translation surfaces with constant Gauss curvature, centroaffine translation surfaces with constant Pick invariant, and centroaffine translation surfaces with||T||2=constant in R3. On the other hand, by solving certern partial differential equations we obtain some classification results for linear Weingarten centroaffine ruled surfaces in3-affine space R3and prove that a nondegenerate centroaffine ruled surface is centroaffine minimal if and only if its Gauss curvature is constant.
For centroaffine immersions of codimension two x:Mn→Rn+2, there always are two different research methods. Nomize and Sasaki defined a pre-normalized Blaschke vector field ξ using the conditiontrh{T(X,Y)+h(SX,Y)}=0, where h is the centroaffine basic form, S is the Weingarten operator, T is centroaffine2-form. On the other hand, Professor Liu has studied centroaffine immersions of codimension two using a new method to define the metric g, and chose Δgx as the second transversal vector fields, where Δg is the Laplacian of metric g. When we study centroaffine immersions of codimension two, ξ and Δgx all can be think as the second transversal vector field, and the relations and differences between them are the questions we must consider. In the fourth part of this paper, we calculate the basic equations and invariants centroaffine immersions of codimension two using moving frames. Depending on these equations and invariants we get the relation between these two second transversal vector fields, that is,ξ=1/nΔgx-H/2x, where H is centroaffine mean curvature. Then we compute the first and also second variational formulas of centroaffine volume integral using Δgx as the second transversal vector field and then define the minimal centroaffine immersions of codimension two, which is consistent to the minimal centroaffine immersions of codimension two using ξ as the second transversal vector field. As some examples, we proof that homogeneous surfaces with vanishing Pick invariant in R4are centroaffine minimal.
In this paper, we introduce the history of Differetial Geometry, especially about Affine Differetial Geometry (ADG) and Centroaffine Differetial Geometry in the first part. In the second part we describe some knowledge in Differetial Geometry, which is the basic of Centroaffine Differetial Geometry. Then in the third part we talk about centroaffine hypersurfaces. Mainly, we get some results about centroaffine translation surfaces and centroaffine ruled surfaces in R3. In the last part we introduce the centroaffine subminafolds of codimention two. Here we compute the first and the second formulas of area variantion and get the relation between different centroaffine normalizations.
引文
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