In this paper, we mainly investigate curvature properties and harmonicity of invariant vector fields on the four-dimensional Oscillator groups endowed with three left-invariant pseudo-Riemannian metrics of signature (2,2). We determine all harmonic vector fields, vector fields which define harmonic maps and the vector fields which are critical points for the energy functional restricted to vector fields of the same length.

中文翻译：

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具有中性信号的振子群上矢量场的调和性

在本文中，我们主要研究具有三个签名的左不变伪黎曼度量的二维振子群的不变矢量场的曲率性质和调和性（2,2）。我们确定所有谐波矢量场，定义谐波图的矢量场和作为能量函数的临界点的矢量场，这些能量函数仅限于相同长度的矢量场。