#### Vol. 317, No. 1, 2022

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Genuine infinitesimal bendings of submanifolds

### Marcos Dajczer and Miguel Ibieta Jimenez

Vol. 317 (2022), No. 1, 119–141
##### Abstract

A basic question in submanifold theory is whether a given isometric immersion $f:{M}^{n}\to {ℝ}^{n+p}$ of a Riemannian manifold of dimension $n\ge 3$ into Euclidean space with low codimension admits, locally or globally, a genuine infinitesimal bending. That is, if there exists a genuine smooth variation of $f$ by immersions that are isometric up to the first order. Until now only the hypersurface case $p=1$ is well understood. We show that a strong necessary local condition to admit such a bending is the submanifold to be ruled and give a lower bound for the dimension of the rulings. In the global case, we describe the situation of compact submanifolds of dimension $n\ge 5$ in codimension $p=2$.

##### Keywords
Euclidean submanifolds, infinitesimal bendings
##### Mathematical Subject Classification
Primary: 53A07, 53B25, 53C40