Euclidean hypersurfaces with genuine deformations in codimension two

TL;DR

This paper classifies rank-two hypersurfaces in Euclidean space that admit genuine isometric deformations into higher codimension, providing a comprehensive understanding of their deformation behavior.

Contribution

It offers a complete classification of hypersurfaces with genuine deformations in codimension two, expanding the understanding of their geometric properties.

Findings

Identifies conditions under which hypersurfaces admit genuine deformations.

Provides a classification of such hypersurfaces in Euclidean space.

Clarifies the nature of isometric deformations in higher codimension.

Abstract

We classify hypersurfaces of rank two of Euclidean space $\R^{n+1}$ that admit genuine isometric deformations in $\R^{n+2}$. That an isometric immersion $\hat f\colon\,M^n\to\R^{n+2}$ is a genuine isometric deformation of a hypersurface $f\colon\, M^n\to\R^{n+1}$ means that $\hat f$ is nowhere a composition $\hat f=\hat F\circ f$, where $\hat F\colon\,V\subset \R^{n+1}\to\R^{n+2}$ is an isometric immersion of an open subset $V$ containing $f(M)$.