Genuine deformations of Euclidean hypersurfaces in higher codimensions I

Abstract

Sbrana and Cartan gave local classifications for the set of Euclidean hypersurfaces $M^n\subseteq {\mathbb{R}}^{n+1}$ which admit another genuine isometric immersions in ${\mathbb{R}}^{n+1}$ for $n\ge 3$. The main goal of this paper is to extend their classification to higher codimensions. Our main result is a complete description of the moduli space of genuine deformations of generic hypersurfaces of rank $(p+1)$ in ${\mathbb{R}}^{n+p}$ for $p\le n-2$. As a consequence, we obtain an analogous classification to the ones given by Sbrana and Cartan providing all local isometric immersions in ${\mathbb{R}}^{n+2}$ of a generic hypersurface $M^n\subseteq {\mathbb{R}}^{n+1}$ for $n\ge 4$. We also show how the techniques developed here can be used to study conformally flat Euclidean submanifolds.

arXiv comment27 pages