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Let x : M n−1 → R n (n ≥ 4) 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. We denote the Laguerre scalar curvature by R and the trace-free Laguerre tensor byL := L − 1 n−1 tr(L)g. In this paper, we prove a local classification result under the assumption of parallel Laguerre form anddoi:10.4134/bkms.b150386 fatcat:kpxbmxlhavehnitray3tc3kaze