Preprint C1448/2019
Genuine infinitesimal bendings of Euclidean submanifolds
Miguel Ibieta Jimenez
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.