Gauss maps of harmonic and minimal great circle fibrations

Pub Date : 2023-02-13 DOI:10.1007/s10455-023-09886-0
Ioannis Fourtzis, Michael Markellos, Andreas Savas-Halilaj
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Abstract

We investigate Gauss maps associated to great circle fibrations of the euclidean unit 3-sphere \(\mathbb {S}^3\). 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 on \(\mathbb {S}^3\), whose integral curves are great circles, is a Hopf vector field.

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调和和最小大圆振动的高斯图
我们研究了与欧氏单位3-球体(\mathbb{S}^3\)的大圆纤维化有关的高斯映射。我们证明了与这种fibration相关的高斯映射是调和的,分别是极小的,当且仅当产生大圆叶理的单位向量场是调和的、分别是最小的。这些结果可以看作是Ruh和Vilms关于欧氏空间中极小子流形的高斯映射的调和性的经典定理的类似物。此外,我们证明了积分曲线为大圆的\(\mathbb{S}^3\)上的调和或最小单位向量场是Hopf向量场。
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