仿射空间中的若干曲线和曲面
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摘要
仿射空间An的仿射几何是讨论图形在仿射变换下的不变量和不变性质的一个几何分支。设A3是具有半Euclidean内积λ=λ1x1y1+λ2x2y2+λ3x3y3的仿射空间,其中λ1=±1,λ2=±1,λ3=±1,x=(x1,x2,x3),y-(y1,y2,y3)∈A3则仿射空间A3称为半Euclidean空间。当λ1=λ2=λ3=1时,半Euclidean空间A3是传统的3维Euclidean空间E3.当λ1=λ2=1,λ3=-1时,半Euclidean空间A3是3维Minkowski空间E13(或Lorentzian空间L3)。对Lorentzian空间的研究为Einstein创建和发展广义相对论提供了有效的数学工具。反过来,广义相对论的发展也大大促进了对Lorentzian空间的研究。从而在半Euclidean空间中研究曲线和曲面是非常有意义的。
本文主要在三维仿射空间A3中讨论了曲线及曲面。对于平面二次曲线,Dong-Soo Kim和Young-Ho Kim讨论了Euclidean空间中平面上椭圆和双曲线的性质。而在本论文第三章的曲线部分中,首先使用支函数h和曲率函数κ的关系刻画了二维Euclidean空间中抛物线的性质,根据这些性质得出当平面曲线的支函数h和曲率函数κ满足条件κ=-p2/8h-3时,该曲线是一条焦点在原点的抛物线。接下来根据同一条平面曲线在Euclidean空间中和在仿射空间中的不同结构方程,建立了仿射曲率κ1与欧氏曲率κ的关系,即
并由仿射曲率κ1为常数时,平面曲线为二次曲线这一特性讨论了Euclidean空间中的曲线,这进一步验证了二次曲线的支函数h和曲率函数κ的关系。最后根据曲线的结构方程讨论了在仿射空间中空间曲线的不变量,得出三维仿射空间中的两个不变量κ2τ(ds)6和与环绕空间的度量选取无关。
在第四章的曲面部分中,首先讨论的是乘积极小曲面S在三维Euclidean空间及三维Minkowski空间中的分类。由于局部的可将曲面r(x,y)看成一个图r(x,y)=(x,y,z(x,y)),这样乘积曲面S的表达式可以写为z=f(x)g(y)、 y=f(x)g(z)或x=f(y)g(z)当具有以上形式的乘积曲面极小时,本文根据相关的微分方程给出了乘积曲面详细的分类。由于Minkowski空间中的旋转可以绕三类旋转轴进行,即类空的,类时的和类光的,则Minkowski空间中的旋转对应着三种类型的旋转矩阵,同一条轮廓曲线在不同的旋转矩阵作用下可得到不同的旋转曲面。本文在第四章的第二部分考虑三维Minkowski空间中旋转曲面的平均曲率H为给定函数时,不同类型的旋转曲面的分类。在三维仿射空间中,直纹曲面S的表达式为x(u,v)=a(u)+vb(u)本文在第四章的第三部分讨论了线性Weingarten中心仿射直纹曲面的分类问题：当具有以上形式的直纹曲面极小时,本文根据结构方程和相关的偏微分方程给出了详细的分类,并得出非退化的中心仿射直纹曲面极小的充分必要条件是其Guass曲率是常数。如果曲面S的高斯曲率K和平均曲率H满足线性关系λ1K+λ2H=λ,其中λ1,λ2,λ是常数,则此曲面称为线性Weingarten曲面。显然,常高斯曲率的曲面和常平均曲率的曲面都是线性Weingarten曲面。本文在给出常高斯曲率的中心仿射直纹曲面和常平均曲率的中心仿射直纹曲面的分类后,又进一步对线性Weingarten中心仿射直纹曲面进行了详细的分类。
Affine geometry of affine space A" is a banch of talking affine invariant and affine invariance. Let A3be3-dimensional space with the semi Euclidean inner productλ=λ1x1y1+λ2x2y2+λ3x3y3, where λ1=±1, λ2=±1, λ3=±1and x=(x1,x2,x2), y=(y1,y2,y3)∈A3, this space is called semi Euclidean space. When λ1=λ2=λ3=1, this semi Euclidean space A3is the3-dimensional Euclidean space E3. When λ1=λ2=1and λ3=-1, this semi Euclidean space A3is the3-dimensional Minkowski space E3(or Lorentzian space L3). It is the Lorentzian space that provides with the effective mathematical method to Einstein who founded and extended the theory of relativity. Also, the theory of relativity accelerates the study of Lorentzian space. So it is useful to study curves and surfaces in the semi Euclidean space.
In this paper, we will study curves and surfaces in3-dimensional affine space. A characterization of the ellipse and hyperbola had been studied by Dong-Soo Kim and Young-Ho Kim. Here in the section of curves, firstly, we will study a characterization of parabola in2-demensional Euclidean space, using the support function h and the curvature k of the parabola. We can get that if the support function and the curvature of a curve satisfy the equation κ=-p2/8h3, the curve is a parabola whose focus is at the origin. Next we get that the relationship between the affine curvature κ1and the Euclidean curvature k for a curve in2-dimensional space is
As we known, a curve is a plane quadratic curve if and only if the affine curvature is a constant. Using this conclution we will characterize plane curves in the Euclidean space. Thirdly, considering a curve as the curve in3-dimensional Euclidean space and also in3-dimensional Minkowski space, we will get some invariants of the curve, such that κ2τ(ds)6and so on, which are independent of the choice of inner products.
In the section of surfaces, there have three parts. The first part will study factorable minmal surfaces in3-dimensional Euclidean space and3-dimensional Minkowski space. As a surface can be regarded as a graph locally, tbe factorable surface can be written as z=f(x)g(y), y=f(x)g(z) or x=f(y)g(z),
By solving equation, we class the factorable minmal surfaces. In the second part, we will study surfaces of revolution with prescribed mean curvature in3-dimensional Minkowski space. Since there are three kinds of rotations in3-dimensional Minkowski space, we have three kinds of rotation matrices respect to spacelike, timelike and lightlike rotation axes. In this paper, we will class these rotation surfaces with the prescribed mean curvature by solving some differential equation. In the last part, by solving certain partial differential equations we obtain some classification results for linear Weingarten centroaffine ruled surfaces in3-dimensional affine space and prove that a non-degenerate centroaffine ruled surface is centroaffine minimal if and only if its Guass curvature is constant.
本文主要在三维仿射空间A3中讨论了曲线及曲面。对于平面二次曲线,Dong-Soo Kim和Young-Ho Kim讨论了Euclidean空间中平面上椭圆和双曲线的性质。而在本论文第三章的曲线部分中,首先使用支函数h和曲率函数κ的关系刻画了二维Euclidean空间中抛物线的性质,根据这些性质得出当平面曲线的支函数h和曲率函数κ满足条件κ=-p2/8h-3时,该曲线是一条焦点在原点的抛物线。接下来根据同一条平面曲线在Euclidean空间中和在仿射空间中的不同结构方程,建立了仿射曲率κ1与欧氏曲率κ的关系,即
并由仿射曲率κ1为常数时,平面曲线为二次曲线这一特性讨论了Euclidean空间中的曲线,这进一步验证了二次曲线的支函数h和曲率函数κ的关系。最后根据曲线的结构方程讨论了在仿射空间中空间曲线的不变量,得出三维仿射空间中的两个不变量κ2τ(ds)6和与环绕空间的度量选取无关。
在第四章的曲面部分中,首先讨论的是乘积极小曲面S在三维Euclidean空间及三维Minkowski空间中的分类。由于局部的可将曲面r(x,y)看成一个图r(x,y)=(x,y,z(x,y)),这样乘积曲面S的表达式可以写为z=f(x)g(y)、 y=f(x)g(z)或x=f(y)g(z)当具有以上形式的乘积曲面极小时,本文根据相关的微分方程给出了乘积曲面详细的分类。由于Minkowski空间中的旋转可以绕三类旋转轴进行,即类空的,类时的和类光的,则Minkowski空间中的旋转对应着三种类型的旋转矩阵,同一条轮廓曲线在不同的旋转矩阵作用下可得到不同的旋转曲面。本文在第四章的第二部分考虑三维Minkowski空间中旋转曲面的平均曲率H为给定函数时,不同类型的旋转曲面的分类。在三维仿射空间中,直纹曲面S的表达式为x(u,v)=a(u)+vb(u)本文在第四章的第三部分讨论了线性Weingarten中心仿射直纹曲面的分类问题：当具有以上形式的直纹曲面极小时,本文根据结构方程和相关的偏微分方程给出了详细的分类,并得出非退化的中心仿射直纹曲面极小的充分必要条件是其Guass曲率是常数。如果曲面S的高斯曲率K和平均曲率H满足线性关系λ1K+λ2H=λ,其中λ1,λ2,λ是常数,则此曲面称为线性Weingarten曲面。显然,常高斯曲率的曲面和常平均曲率的曲面都是线性Weingarten曲面。本文在给出常高斯曲率的中心仿射直纹曲面和常平均曲率的中心仿射直纹曲面的分类后,又进一步对线性Weingarten中心仿射直纹曲面进行了详细的分类。
Affine geometry of affine space A" is a banch of talking affine invariant and affine invariance. Let A3be3-dimensional space with the semi Euclidean inner product
In this paper, we will study curves and surfaces in3-dimensional affine space. A characterization of the ellipse and hyperbola had been studied by Dong-Soo Kim and Young-Ho Kim. Here in the section of curves, firstly, we will study a characterization of parabola in2-demensional Euclidean space, using the support function h and the curvature k of the parabola. We can get that if the support function and the curvature of a curve satisfy the equation κ=-p2/8h3, the curve is a parabola whose focus is at the origin. Next we get that the relationship between the affine curvature κ1and the Euclidean curvature k for a curve in2-dimensional space is
As we known, a curve is a plane quadratic curve if and only if the affine curvature is a constant. Using this conclution we will characterize plane curves in the Euclidean space. Thirdly, considering a curve as the curve in3-dimensional Euclidean space and also in3-dimensional Minkowski space, we will get some invariants of the curve, such that κ2τ(ds)6and so on, which are independent of the choice of inner products.
In the section of surfaces, there have three parts. The first part will study factorable minmal surfaces in3-dimensional Euclidean space and3-dimensional Minkowski space. As a surface can be regarded as a graph locally, tbe factorable surface can be written as z=f(x)g(y), y=f(x)g(z) or x=f(y)g(z),
By solving equation, we class the factorable minmal surfaces. In the second part, we will study surfaces of revolution with prescribed mean curvature in3-dimensional Minkowski space. Since there are three kinds of rotations in3-dimensional Minkowski space, we have three kinds of rotation matrices respect to spacelike, timelike and lightlike rotation axes. In this paper, we will class these rotation surfaces with the prescribed mean curvature by solving some differential equation. In the last part, by solving certain partial differential equations we obtain some classification results for linear Weingarten centroaffine ruled surfaces in3-dimensional affine space and prove that a non-degenerate centroaffine ruled surface is centroaffine minimal if and only if its Guass curvature is constant.
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