Infinitesimal bendings of complete Euclidean hypersurfaces

Abstract

A local description of the non-flat infinitesimally bendable Euclidean hypersurfaces was recently given by Dajczer and Vlachos (Ann Mat 196:1961–1979, 2017. https://doi.org/10.1007/s10231-017-0641-8). From their classification, it follows that there is an abundance of infinitesimally bendable hypersurfaces that are not isometrically bendable. In this paper we consider the case of complete hypersurfaces \(f:M^n\rightarrow \mathbb {R}^{n+1}\), \(n\ge 4\). If there is no open subset where f is either totally geodesic or a cylinder over an unbounded hypersurface of \(\mathbb {R}^4\), we prove that f 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 field of an isometric bending.