A basic question in submanifold theory is whether a given isometric immersion
of a Riemannian
manifold of dimension
into Euclidean space with low codimension admits, locally or globally, a genuine
infinitesimal bending. That is, if there exists a genuine smooth variation
of by
immersions that are isometric up to the first order. Until now only the hypersurface case
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
in
codimension
.
