On Laguerre form and Laguerre isoparametric hypersurfaces

Let x: M^n-1 → Rⁿ be an umbilical free hypersurface with non-zero principal curvatures. M is called Laguerre isoparametric if it satisfies two conditions, namely, it has vanishing Laguerre form and has constant Lauerre principal curvatures. In this paper, under the condition of having constant Laguerre principal curvatures, we show that the hypersurface is of vanishing Laguerre form if and only if its Laguerre form is parallel with respect to the Levi-Civita connection of its Laguerre metric.