We extend the notion of $r$-minimality of a submanifold in arbitrary codimension to $u$-minimality for a multi-index $u\in\mathbb{N}^q$, where $q$ is the codimension. This approach is based on the analysis on the frame bundle of orthonormal frames of the normal bundle to a submanifold and vector bundles associated with this bundle. The notion of $u$-minimality comes from the variation of $\sigma_u$-symmetric function obtained from the family of shape operators corresponding to all possible bases of the normal bundle. We obtain the variation field, which gives alternative definition of $u$--minimality. Finally, we give some examples of $u$-minimal submanifolds for some choices of $u$.

中文翻译：

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广义最小子流形的框架束方法

对于多索引 $u\in\mathbb{N}^q$，我们将任意余维中子流形的 $r$-极小性的概念扩展到 $u$-极小，其中 $q$ 是余维。这种方法基于对法线束的正交框架的框架束的分析，到子流形和与该束相关联的向量束。$u$-minimality 的概念来自 $\sigma_u$-对称函数的变体，该函数从对应于正规丛的所有可能基的形状算子族中获得。我们获得了变异场，它给出了 $u$--minimalality 的替代定义。最后，我们给出一些 $u$-minimal submanifolds 的一些例子，用于 $u$ 的一些选择。