Genuine infinitesimal bendings of submanifolds

Abstract

A basic question in submanifold theory is whether a given isometric immersion $f:M^n\to {\mathbb{R}}^{n+p}$ of a Riemannian manifold of dimension $n\ge 3$ 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 $n\ge 5$ in codimension $p=2$.