三维Minkowski空间中的平移曲面

Translation surfaces in 3-D Minkowski space

讨论三维Minkowski空间中的平移曲面。曲面的性质主要取决于高斯曲率和平均曲率,所以研究曲面的高斯曲率和平均曲率之间的关系,也就是曲面的Weingarten型有着重要的意义。在三维Minkowski空间中,存在类空、类时、和类光3种向量,选取这3种向量中的任意2种作为2个平移方向,可以将平移曲面分为6类。在伪正交标架下,选取一种新的度量形式,对沿类光和类空方向平移的平移Weingarten曲面进行了研究。首先,根据微分几何中的基本知识,得到了该种度量形式下的平移曲面的第1、第2基本形式以及高斯曲率和平均曲率;然后,主要利用高斯曲率和平均曲率之间的线性关系和平方关系,得到了这类平移曲面的分类定理。

This paper considers translation surfaces in 3-D Minkowski space.The nature of surface mainly depends on Gaussian curvature and mean curvature,and therefore,it is of significance to investigate the relation between Gaussian curvature and mean curvature of surface,i.e., Weingarten surface.There are three kinds of vectors in the 3-D Minkowski space,i.e.,space-like,time-like and light-like vectors among which choosing any two vectors as the directions of translation will divide the translation surfaces into six types.A new metric form is chosen to study the Weingarten translation surfaces which are translating in the lightlike direction and spacelike direction in a pseudoorthogonal frame.Then,the first and second fundamental forms,Gaussian curvature and mean curvature of the surfaces are directly calculated according to the principles of differential geometry.It follows that some theorems of classification of those translation surfaces are given mainly by virtue of the linear and square relationships between the Gaussian curvature and the mean curvature.