论文标题
谐波和最小圆圈的高斯图
Gauss maps of harmonic and minimal great circle fibrations
论文作者
论文摘要
我们研究了与$ s^3 $的大圆纤维相关的高斯地图。我们表明，当且仅当单位矢量场产生大圆形叶子是谐波（分别最小）时，相关的高斯映射与这样的振动是谐波（分别最小）。这些结果可以看作是RUH和VILMS经典定理的类似物，涉及欧几里得空间中最小的亚曼福尔德的高斯图的谐波。此外，我们证明，具有Great Circle积分曲线的谐波或最小单元向量字段是Hopf Vector Field。
We investigate Gauss maps associated to great circle fibrations of $S^3$. We show that the associated Gauss map to such a fibration is harmonic (respectively minimal) if and only if the unit vector field generating the great circle foliation is harmonic (respectively minimal). These results can be viewed as analogues of the classical theorem of Ruh and Vilms about the harmonicity of the Gauss map of a minimal submanifold in the euclidean space. Moreover, we prove that a harmonic or minimal unit vector field in $S^3$ with great circle integral curves is a Hopf vector field.
