In this paper, we study the biharmonic submanifolds of Riemannian manifolds endowed with metallic and complex metallic structures. In case of both the structures, we obtain the necessary and sufficient conditions for a submanifold to be biharmonic. Particularly, we find the estimates for mean curvature of Lagrangian and complex surfaces.

中文翻译：

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金属黎曼流形中的双谐波子流形

在本文中，我们研究了具有金属和复杂金属结构的黎曼流形的双调和子流形。在这两种结构的情况下，我们获得了子流形成为双谐波的充分必要条件。特别是，我们找到了对拉格朗日和复曲面的平均曲率的估计。