A hypersurface is said to be totally biharmonic if all its geodesics are biharmonic curves in the ambient space. We prove that a totally biharmonic hypersurface into a space form is an isoparametric biharmonic hypersurface, which allows us to give the full classification of totally biharmonic hypersurfaces in these spaces. Moreover, restricting ourselves to the 3-dimensional case, we show that totally biharmonic surfaces into Bianchi–Cartan–Vranceanu spaces are isoparametric surfaces and we give their full classification. In particular, we show that, leaving aside surfaces in the 3-dimensional sphere, the only nontrivial example of a totally biharmonic surface appears in the product space 𝕊2(ρ) × ℝ.

中文翻译：

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空间形式和 3 维 BCV 空间中的完全双调和超曲面

如果超曲面的所有测地线都是环境空间中的双调和曲线，则称超曲面是完全双调和的。我们证明了空间形式的全双调超曲面是等参双调和超曲面，这使我们能够对这些空间中的全双调超曲面进行完整分类。此外，将自己限制在 3 维情况下，我们表明进入 Bianchi-Cartan-Vranceanu 空间的完全双调和曲面是等参曲面，我们给出了它们的完整分类。特别是，我们表明，撇开 3 维球体中的表面，完全双调和表面的唯一重要示例出现在乘积空间中𝕊2(ρ) × ℝ.