论文标题
扩展三维Riemannian Lie组的谐波矢量场
Harmonic vector fields on extended 3-dimensional Riemannian Lie groups
论文作者
论文摘要
鉴于两个Riemannian流形$（B，G_B）$和$（F，G_F）$，我们为Riemannian Warped Product $ b \ times_ff $上的矢量字段提供了谐调条件，使用$ f：b \ longrightArrow] 0，+\ flongrightArrow] 0，+\ iffty [$，使用特征性变异条件。然后，我们将其应用于$ b = \ mathbb {r} $，$ f $是配备有剩余数量的三维连接的Riemannian Lie Group $ G $，以确定$ \ Mathbb {r} \ times_fg $的谐波向量字段。我们在$ g $上提供了谐波矢量字段的示例，这些示例不是剩余不变的，并在$ \ mathbb {r} \ times_fg $上确定谐波向量字段。我们以$ \ mathbb {r} \ times_fg $为谐波图上的矢量字段的一些示例。
Given two Riemannian manifolds $(B,g_B)$ and $(F,g_F)$, we give harmonicity conditions for vector fields on the Riemannian warped product $B\times_fF$, with $f:B \longrightarrow ]0,+\infty[$, using a characteristic variational condition. Then, we apply this to the case $B=\mathbb{R}$ and $F$ is a three-dimensional connected Riemannian Lie group $G$ equipped with a left-invariant metric, to determine harmonic vector fields on $\mathbb{R}\times_fG$. We give examples of harmonic vector fields on $G$ which are not left-invariant and determine harmonic vector fields on $\mathbb{R}\times_fG$. We conclude with some examples of vector fields on $\mathbb{R}\times_fG$ which are harmonic maps.
