  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
$p=2$.
