Compatibility of Gauss maps with metrics

TL;DR

This paper establishes precise conditions under which a smooth map into a sphere can serve as the Gauss map of an isometric immersion, advancing understanding of geometric compatibility in Riemannian geometry.

Contribution

It provides necessary and sufficient conditions for a map into the sphere to be realizable as a Gauss map of an isometric immersion, including the case of general codimension.

Findings

Characterization of Gauss maps for isometric immersions

Conditions for maps into spheres to be Gauss maps

Discussion on general codimension cases

Abstract

We give necessary and sufficient conditions on a smooth local map of a Riemannian manifold $M^m$ into the sphere $S^m$ to be the Gauss map of an isometric immersion $u:M^m \to R^n$, $n=m+1$. We briefly discuss the case of general $n$ as well