Classification of hypersurfaces with parallel Laguerre second fundamental form in \(\mathbb R^n\) (Q960868)

Let \(U\mathbb{R}^n\) be the unit tangent bundle over \(\mathbb{R}^n\). A diffeomorphism \(\phi:U\mathbb{R}^n\to U\mathbb{R}^n\) is called a Laguerre transformation if it takes oriented spheres to oriented spheres, oriented hyperplanes to oriented hyperplanes, preserving the tangential distance of any two spheres. Let \(x: M\to\mathbb{R}^n\) be an umbilical free hypersurface with non-zero principal curvatures. In the paper all oriented hypersurfaces with a parallel Laguerre second fundamental form are classified.