Euclidean Hypersurfaces with Genuine Conformal Deformations in Codimension Two

Abstract

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