Genuine deformations of Euclidean hypersurfaces in higher codimensions I

TL;DR

This paper extends the classification of Euclidean hypersurfaces with genuine isometric deformations from codimension one to higher codimensions, providing a comprehensive description of their moduli space and implications for conformally flat submanifolds.

Contribution

It offers a complete classification of genuine deformations of generic hypersurfaces in higher codimensions, generalizing classical results by Sbrana and Cartan.

Findings

Complete description of the moduli space of genuine deformations

Classification of local isometric immersions in higher codimensions

Application of techniques to conformally flat Euclidean submanifolds

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 $n\geq 3$. 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 $p\leq n-2$. 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 $n\geq 4$. We also show how the techniques developed here can be used to study conformally flat Euclidean submanifolds.