Singular genuine rigidity

TL;DR

This paper generalizes the concept of genuine rigidity to include mild singularities, leading to new global rigidity results and unifying classical theorems with a simpler, more natural theory.

Contribution

It introduces a singular extension of genuine rigidity, unifies Sacksteder and Dajczer-Gromoll theorems, and simplifies the theory by removing previous technical codimension constraints.

Findings

Any compact n-dimensional submanifold of R^{n+p} is singularly genuinely rigid in R^{n+q} for q < min{5,n} - p

The singular theory simplifies and becomes more natural than the regular case

All previous codimension assumptions are removed in the singular setting

Abstract

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.