# Genuine infinitesimal bendings of submanifolds

## TL;DR

This paper investigates the conditions under which submanifolds in Euclidean space admit genuine infinitesimal bendings, providing new insights into the geometric structure and restrictions for such deformations.

## Contribution

It establishes a necessary local condition that submanifolds must be ruled to admit infinitesimal bendings and describes the global case for certain compact submanifolds.

## Key findings

- A submanifold must be ruled to admit a genuine infinitesimal bending.
- Provides a lower bound for the dimension of rulings.
- Describes the global situation for compact submanifolds in specific dimensions and codimensions.

## Abstract

A basic question in submanifold theory is whether a given isometric immersion $f\colon M^n\to\R^{n+p}$ of a Riemannian manifold of dimension $n\geq 3$ into Euclidean space with low codimension $p$ 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 $p=1$ was 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 $n\geq 5$ in codimension $p=2$.