We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer-Gromoll theorems by showing that any compact $n$-dimensional submanifold of ${\mathbb R}^{n+p}$ is singularly genuinely rigid in ${\mathbb R}^{n+q}$, for any $q < \min\{5,n\} - p$. Unexpectedly, the singular theory becomes much simpler and natural than the regular one, even though all technical codimension assumptions, needed in the regular case, are removed.

中文翻译：

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单一的真正刚性

我们通过允许温和奇点来扩展子流形的真正刚性的概念，主要是为了获得新的全局刚性结果并统一已知的结果。作为结果之一，我们通过证明 ${\mathbb R}^{n+p}$ 的任何紧凑 $n$ 维子流形在 ${\ mathbb R}^{n+q}$，对于任何 $q < \min\{5,n\} - p$。出乎意料的是，奇异理论变得比常规理论简单和自然得多，即使在常规情况下所需的所有技术共维假设都被删除了。