Laguerre hypersurfaces with definite Laguerre tensor

x : Mn-1 ? Rn be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. In this paper, we will study the Laguerre hypersurface with parallel Laguerre form and nonnegative or non-positive Laguerre tensor.