Euclidean Hypersurfaces with Genuine Conformal Deformations in Codimension Two

In this paper we classify Euclidean hypersurfaces $f\colon M^n \rightarrow $\mathbb{R}$^{n+1}$withaprincipalcurvatureofmultiplicity$n-2$ that admit a genuine conformal deformation $\tilde{f}\colon M^n \rightarrow $\mathbb{R}$^{n+2}$\mathrm{.}That$\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 $\tilde{f}{\mathrm{\mid }}_U$ is a composition $\tilde{f}{\mathrm{\mid }}_U=h\circ f{\mathrm{\mid }}_U$ of $f{\mathrm{\mid }}_U$ with a conformal immersion $h\text{\hspace{0.17em}⁣}:V\to {\mathbb{R}}^{n+2}$ of an open subset $V\subset $\mathbb{R}$^{n+1}$containing$f(U)$.