Classification of codimension two deformations of rank two Riemannian manifolds

TL;DR

This paper characterizes the local deformation space of rank two Euclidean submanifolds in codimension two, providing explicit representations, identifying genuinely deformable examples, and exploring their isometric immersions as hypersurfaces.

Contribution

It offers a complete description of the moduli space of deformations for these submanifolds and introduces the first known genuinely deformable examples in this setting.

Findings

Explicit representation of rank two Euclidean submanifolds

Identification of genuinely deformable submanifolds

Conditions for isometric immersion as Euclidean hypersurfaces

Abstract

The purpose of this work is to close the local deformation problem of rank two Euclidean submanifolds in codimension two by describing their moduli space of deformations. In the process, we provide an explicit simple representation of these submanifolds, a result of independent interest by its applications. We also determine which deformations are genuine and honest, allowing us to find the first known examples of honestly locally deformable rank two submanifolds in codimension two. In addition, we study which of these submanifolds admit isometric immersions as Euclidean hypersurfaces, a property that gives rise to several applications to the Sbrana-Cartan theory of deformable Euclidean hypersurfaces.