We first prove that, unlike the biharmonic case, there exist triharmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classification of triharmonic curves in surfaces with constant Gaussian curvature. Next, restricting to curves in a 3-dimensional Riemannian manifold, we study the family of triharmonic curves with constant curvature, showing that they are Frenet helices. In the last part, we give the full classification of triharmonic Frenet helices in space forms and in Bianchi–Cartan–Vranceanu spaces.

中文翻译：

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三维齐次空间中的三谐曲线

我们首先证明，与双调和情况不同，在任意维数的合适黎曼流形中存在曲率不等的三调和曲线。然后我们给出了具有恒定高斯曲率的曲面中三谐曲线的完整分类。接下来，限制在 3 维黎曼流形中的曲线，我们研究了具有恒定曲率的三谐曲线族，表明它们是 Frenet 螺旋。在最后一部分，我们给出了空间形式和 Bianchi-Cartan-Vranceanu 空间中三谐 Frenet 螺旋的完整分类。