On Laguerre Isotropic Hypersurfaces

We study Laguerre isotropic hypersurfaces in the Euclidean space, which are hypersurfaces whose Laguerre form is zero and the eigenvalues of the Laguerre tensor are constant and equal to $\lambda \ge 0$. We prove a rigidity theorem for the L-isotropic hypersurfaces parametrized by lines of curvature. Moreover, we study the hypersurfaces that are L-isotropic and L-isoparametric simultaneously and we show that for such a hypersurface $\lambda =0$. We obtain necessary conditions for the existence of L-isotropic hypersurfaces with $\lambda >0$ and we prove that a certain function, determined by the radii of curvature of the hypersurface, is bounded above by $1\mathrm{/}2\lambda$.