**Keywords:**Euclidean submanifolds | infinitesimal bendings | complete hypersurfaces.

A notion of bending of an isometric immersion

$f\colon M^n\to\R^{n+p}$ is associated to

smooth variations of $f$ by immersions that are isometric up to the first

order. More precisely, an infinitesimal bending of $f$ is the variational vector field

associated to such variation.

A very basic question in submanifold theory is

whether a given isometric immersion $f\colon M^n\to\R^{n+p}$

with low codimension admits, locally or globally, a genuine

infinitesimal bending. That is, if $f$ admits an infinitesimal bending that is not

determined by an infinitesimal bending of a submanifold of larger dimension that contains $f(M)$.

We show that a strong necessary local condition to admit such a bending

is the submanifold to be ruled and we give a lower bound to the dimension

of the rulings. In the global case, we describe the situation for infinitesimal bendings of

compact submanifolds with dimension at least five in codimension two.

In the codimension one case, a local description of the non-flat infinitesimally bendable Euclidean

hyeprsurfaces was recently given by

Dajczer and Vlachos. From their classification, it follows that this class is much larger than the class

of isometrically bendable ones.

In this work we also prove that a complete Euclidean hypersurface

$f\colon M^n\to\R^{n+1}$, $n\geq 4$, having no open subset where $f$ is totally geodesic

or a cylinder over an unbounded hypersurface of $\R^4$, is infinitesimally bendable only along ruled strips.

In particular, if the hypersurface is simply connected, this implies that any infinitesimal bending of $f$

is the variational vector field of an isometric bending, in contrast with the local case.