Genuine deformations of submanifolds II:the conformal case

TL;DR

This paper extends the concept of genuine deformations to conformal submanifolds, characterizing their geometric structure and demonstrating the unifying nature of this approach through various applications.

Contribution

It introduces the notion of genuine conformal deformations, generalizing previous isometric cases, and provides a detailed geometric description and applications.

Findings

Characterization of submanifolds admitting genuine conformal deformations

Identification of geometric structures associated with these deformations

Applications illustrating the unifying framework of the concept

Abstract

We extend to the conformal realm the concept of genuine deformations of submanifolds, introduced by Dajczer and the first author for the isometric case. Analogously to that case, we call a conformal deformation of a submanifold $M^n$ genuine if no open subset of $M^n$ can be included as a submanifold of a higher dimensional conformally deformable submanifold in such a way that the conformal deformation of the former is induced by a conformal deformation of the latter. We describe the geometric structure of a submanifold that admits a genuine conformal deformation and give several applications showing the unifying character of this concept.