东北大学 理学院, 辽宁 沈阳 110819
收稿日期:2017-10-17
基金项目:国家自然科学基金资助项目(11801065,11371080)。
作者简介:钱金花(1979 -),女,河北唐山人,东北大学副教授。
摘要:在三维闵可夫斯基(Minkowski)空间中定义了以类时曲线为脊线的圆纹(canal)曲面, 并对温加顿(Weingarten)圆纹曲面进行了分类.与三维欧氏空间类似, 首先以类时曲线的伏雷内(Frenet)标架为基础, 结合圆纹曲面的几何定义, 得到了伪正交标架下以类时曲线为脊线的圆纹曲面的参数方程.然后, 建立此类圆纹曲面的基本理论, 包括第一、第二基本量, 高斯曲率和平均曲率等.在此基础上, 得到了高斯曲率和平均曲率之间的关系, 并对Weingarten圆纹曲面进行了详细的讨论.得到了三维Minkowski空间中以类时曲线为脊线的Weingarten圆纹曲面是管道曲面或者旋转曲面的结论.
关键词:Minkowski空间圆纹曲面Weingarten曲面高斯曲率平均曲率
Canal Surfaces in 3D Minkowski Space
QIAN Jin-hua, FU Xue-shan
School of Sciences, Northeastern University, Shenyang 110819, China
Corresponding author: QIAN Jin-hua, E-mail: qianjinhua@mail.neu.edu.cn
Abstract: The canal surfaces with time-like center curves in 3D Minkowski space were defined and the Weingarten canal surfaces were classified. Similar to the studying method for surfaces in Euclidean space, at first, the parametric equation of canal surfaces under pseudo orthogonal frame was built according to the Frenet frame of time-like curves and the geometric definition of canal surfaces, then the basic theories were obtained which include two fundamental quantities, the Gaussian curvature and mean curvature and so on. Using basic theories, the relationship between the Gaussian curvature and the mean curvature were found and the Weingarten canal surfaces were studied explicitly. The conclusion was achieved that a canal surface is a Weingarten surface if and only if it is a tube or a revolution surface.
Key words: Minkowski spacecanal surfacesWeingarten surfacesGaussian curvaturemean curvature
1850年Monge将由单参数球面族沿脊线运动生成的包络面定义为圆纹曲面[1].本文将欧氏空间中的圆纹曲面推广到三维Minkowski空间.将由单参数伪球面族S12沿脊线运动生成的包络面定义为圆纹曲面, 并对Weingarten圆纹曲面进行了分类.本文主要讨论了以类时曲线为脊线的圆纹曲面, 用类似的方法可以讨论以类空曲线和类光曲线为脊线的圆纹曲面的性质.
1 预备知识设E13是三维Minkowski空间, 其中的内积定义为
设c是E13中任意一条正则曲线.若曲线c的切向量为类空向量(类时向量、类光向量), 则称c为类空曲线(类时曲线、类光曲线).类似地, 设x=x(u, v)是E13中的任意正则曲面, 若曲面x的法向量为类空向量(类时向量、类光向量), 则称曲面x为类时曲面(类空曲面、类光曲面)[3].
引理1[4]??设c(s)是以s为弧长参数的类时曲线, 则其满足如下Frenet公式:
(1) |
定义1[5]??设p是E13中的一固定点, C>0是常数.则E13中的伪黎曼球定义为
定义2??设S是E13中由伪球族S12沿一条类时脊线c(s)运动所生成的圆纹曲面.则曲面S可表示为
标注1??特别地, 当脊线为直线时, 其Frenet标架可看作正交标架, 圆纹曲面为旋转曲面; 当半径函数为常数时, 圆纹曲面又称为管道曲面[8].
标注2??除非特殊说明, 本文所讨论的都是E13中以类时曲线为脊线的圆纹曲面, 后续不再赘述.
定义3[9-10]??若曲面的高斯曲率K和平均曲率H满足Φ(K, H)=0, 其中Φ是雅克比函数, 则称其为Weingarten曲面.
2 主要结论根据定义2, 为了方便, 令r′(s)=tanφ, 这里φ=φ(s)为光滑函数, 则曲面S可表示为
(2) |
下面计算圆纹曲面的两个基本量、高斯曲率以及平均曲率.
首先, 对式(2)分别关于s, θ求偏导数, 结合式(1)得
(3) |
(4) |
(5) |
(6) |
定理1??三维Minkowski空间中以类时曲线为脊线的圆纹曲面是类时曲面.
对式(6)分别关于s, θ求偏导数, 可得
(7) |
(8) |
定理2??设S是E13中的圆纹曲面, 则S的高斯曲率K和平均曲率H可表示为
证明??首先, 由定理2, 通过计算得
(9) |
所以有下列两种情况:
1) 当r′=0, Kθ≠0时, S的半径函数r为常数, 此时, S是管道曲面.
2) 当Kθ=0, r′≠0时, 有κ≡0, 此时, S是旋转曲面.
反之, 若S是旋转曲面, 则有κ=0, 代入定理2中的公式, 得
另一方面, 若S是管道曲面, 则半径函数r为常数, 代入定理2中的公式并求导, 得
3 结语本文将三维欧氏空间中的圆纹曲面推广到Minkowski空间.定义了以类时曲线为脊线的圆纹曲面, 并对Weingarten圆纹曲面进行了分类.此项工作开创了Minkowski空间圆纹曲面研究的先例, 为其他类型圆纹曲面的研究奠定了坚实的基础.
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