Euclidean Hypersurfaces with Genuine Conformal Deformations in Codimension Two

TL;DR

This paper classifies Euclidean hypersurfaces with a specific principal curvature property that admit genuine conformal deformations into higher codimension, expanding understanding of conformal geometry in Euclidean spaces.

Contribution

It provides a classification of hypersurfaces with a principal curvature of multiplicity n-2 that can be genuinely conformally deformed into codimension two.

Findings

Classification of hypersurfaces with the given curvature property.

Conditions under which genuine conformal deformations exist.

Insights into the structure of conformal deformations in Euclidean space.

Abstract

In this paper we classify Euclidean hypersurfaces $f\colon M^n \rightarrow \mathbb{R}^{n+1}$ with a principal curvature of multiplicity $n-2$ that admit a genuine conformal deformation $\tilde{f}\colon M^n \rightarrow \mathbb{R}^{n+2}$. That $\tilde{f}\colon M^n \rightarrow \mathbb{R}^{n+2}$ is a genuine conformal deformation of $f$ means that it is a conformal immersion for which there exists no open subset $U \subset M^n$ such that the restriction $\tilde{f}|_U$ is a composition $\tilde f|_U=h\circ f|_U$ of $f|_U$ with a conformal immersion $h\colon V\to \mathbb{R}^{n+2}$ of an open subset $V\subset \mathbb{R}^{n+1}$ containing $f(U)$.