Euclidean Hypersurfaces with Genuine Conformal Deformations in Codimension Two

Abstract

In this paper we classify Euclidean hypersurfaces $f:M^n\to {\mathbb{R}}^{n+1}$ with a principal curvature of multiplicity $n-2$ that admit a genuine conformal deformation $\tilde{f}:M^n\to {\mathbb{R}}^{n+2}$. That $\tilde{f}:M^n\to {\mathbb{R}}^{n+2}$ 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 $h:V\to {\mathbb{R}}^{n+2}$ of an open subset $V\subset {\mathbb{R}}^{n+1}$ containing $f(U)$.

42 pages