四维复欧氏空间单位球面中的一类浸入环面

Torus Immersed in the Unit Sphere in the Four-dimensional Complex Euclidean Space

所研究的对象是源于小波分析滤波器构造理论中提出的几何模型,即一类四维复欧氏空间单位球面中的浸入环面问题,特点是其参数表示中的4个坐标分量函数均为实系数二元多项式。首先根据环面的参数表示得到了多项式系数所满足的约束条件方程组;在此基础上考虑了多项式次数n=1时的情形,得出了此时该环面不可能为全测地浸入的结论;从而以新的研究方法验证了欧氏空间中不存在全测地环面子流形这一事实。

The subject comes from geometric models proposed by the theory of wavelet filter construetion. In other words, it is toms immersed in the unit sphere in the four - dimensional complex Euclidean space, whose four coordinate functions in its parametric expressions are all bivariate polynomials with real coefficients. First, the constraint equations of those coefficients are obtained based on the parametric expressions. Then the case where the degree of the polynomial n = 1 is considered; it is concluded that the toms cannot be totally geodesic, providing a new proof of the fact that it is impossible to find any torus totally geodesic in Euclidean space.