伪欧氏空间中直纹面的性质

Properties of Ruled Surfaces in Pseudo-Euclidean Space

讨论伪欧氏空间中的直纹面。利用活动标架法研究了直纹面的一些性质,包括极小性,全可展性,全测地性和全脐性,给出了直纹面是全可展性的一组充要条件,同时得到,Rnv+1中的k+1维直纹面M是全测地的充要条件是它是极小的且全可展的。特别,若M的生成空间是类空的或类时的,则当k≥2时,M全测地与全脐等价。本文还讨论了Rnv+1中直纹超曲面的Gauss-Kronecker曲率G,当n≥3时,G=0。这与低维情形绝然不同,在R3或R31中只有当直纹面是可展时,高斯曲率才为0。

The （k ＋ 1 ） -dimensional ruled surfaces in pseudo- Euclidean space are studied. By means of moving frame, some properties of ruled surfaces are obtained, including the surfaces being minimal, totally geodesic, totally developableand totally umbilical. Some equivalent conditions in which the ruled surfaces are totally developable are obtained. It is abtained that （k ＋ 1 ） - dimensional ruled surfaces are totally geodesic if and only if they are minimal and totally developable. Specially,when k≥2 and the generating spaces are spacelike or timelike, they are totally geodesic if and only if they are totally umbilical. Finally, Gauss - Kronecker curvature of n - dimensional ruled hypersurface vanishs providing n≥3. But the ruled surfaces in R^3 or R1^3 have vanishing Gauss curvature if and only if they are developable.