Complete hypersurfaces with constant laguerre scalar curvature in .n

Let x: M n.1 . Rn be an umbilical free hypersurface with non-zero principal curvatures. Two basic invariants of M under the Laguerre transformation group of Rn are Laguerre form C and Laguerre tensor L. In this paper, n > 3) complete hypersurface with vanishing Laguerre form and with constant Laguerre scalar curvature R in Rn, denote the trace-free Laguerre tensor by .L.=L.1n.1tr(L) · Id. If L.=L.1n.1tr(L), then M is Laguerre equivalent to a Laguerre isotropic hypersurface; and if supM..L...=(n.1)(n.2).R(n.1)(n.2)(n.3),, M is Laguerre equivalent to the hypersurface .x: H 1 . S n.2 . Rn.