Infinitesimally Moebius bendable hypersurfaces

Abstract

Li, Ma and Wang have provided in [\emph{Deformations of hypersurfaces preserving the M\"obius metric and a reduction theorem}, Adv. Math. 256 (2014), 156--205] a partial classification of the so-called Moebius deformable hypersurfaces, that is, the umbilic-free Euclidean hypersurfaces $f\colon M^n\to \mathbb{R}^{n+1}$ that admit non-trivial deformations preserving the Moebius metric. For $n\geq 5$, the classification was completed by the authors in \cite{JT2}. In this article we obtain an infinitesimal version of that classification. Namely, we introduce the notion of an infinitesimal Moebius variation of an umbilic-free immersion $f\colon M^n\to \mathbb{R}^m$ into Euclidean space as a one-parameter family of immersions $f_t\colon M^n\to \mathbb{R}^m$, with $t\in (-\epsilon, \epsilon)$ and $f_0=f$, such that the Moebius metrics determined by $f_t$ coincide up to the first order. Then we characterize isometric immersions $f\colon M^n\to \mathbb{R}^m$ of arbitrary codimension that admit a non-trivial infinitesimal Moebius variation among those that admit a non-trivial conformal infinitesimal variation, and use such characterization to classify the umbilic-free Euclidean hypersurfaces of dimension $n\geq 5$ that admit non-trivial infinitesimal Moebius variations.