Isometric Euclidean submanifolds with isometric Gauss maps

TL;DR

This paper studies specific isometric immersions of Riemannian manifolds into Euclidean space with codimension two, focusing on deformations that preserve the Gauss map's metric, and provides a local classification for certain cases.

Contribution

It characterizes when such submanifolds are hypersurfaces of deformable hypersurfaces, offering a local classification and parametric description for particular classes.

Findings

Minimal isometric deformations always preserve the third fundamental form.

Non-minimal, non-reducible submanifolds are hypersurfaces of deformable hypersurfaces.

A complete local parametric description is provided for a specific class.

Abstract

We investigate isometric immersions $f\colon M^n\to\R^{n+2}$, $n\geq 3$, of Riemannian manifolds into Euclidean space with codimension two that admit isometric deformations that preserve the metric of the Gauss map. In precise terms, the preservation of the third fundamental form of the submanifold must be ensured throughout the deformation. For minimal isometric deformations of minimal submanifolds this is always the case. Our main result is of a local nature and states that if $f$ is neither minimal nor reducible, then it is a hypersurface of an isometrically deformable hypersurface $F\colon\tilde{M}^{n+1}\to\R^{n+2}$ such that the deformations of $F$ induce those of $f$. Moreover, for a particular class of such submanifolds, a complete local parametric description is provided.