双曲空间运动群Lie子群作用的轨道

双曲空H~3在Weierstrass坐标下的运动群就是Lorentz群G,最近,我们确定了G的各类Lie子群,本文则集中讨论G的2参数Lie予群H在H~3上作用的轨道,主要结果如下:定理1 Lorentz群G的2参数Lie子群H在双曲空间H~3上作用的轨道,或者是圆柱面,或者是非正常曲率的全脐点曲面,反之亦然.定理2 Lorentz群G的每个2参数Lie子群H在双曲空间H~3上作用所得的不同轨道,彼此测地平行.

The Orbits of Lie 2 -subgroups of the Lorents Group G Acting on the Hyperbolic Space

In the coordinates of Weierstrass, the group of motions of the hyperbolic space H3 is isomorphic to the Lorents group G, G and its Lie subgroups act naturally on H3. In this paper, we consider only the two-parameters (Lie) subgroups of G and have the main results as follows:Theorem 1. The orbits of Lie 2 -subgroups of G arc either the cylinders, or the tofally umbilical surfaces of non- positive constant curvature, and conversely.Theorem 2. The orbits of every Lie 2-subgroup H of the Lorents group G are geodesically parallel to one another.