LAGUERRE CHARACTERIZATION OF SOME HYPERSURFACES

Abstract

Abstract. Let x : M n−1 → R n (n ≥ 4) be an umbilical free hyper-surface with non-zero principal curvatures. Then x is associated witha Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and aLaguerre second fundamental form B, which are invariants of x underLaguerre transformation group. We denote the Laguerre scalar curvatureby R and the trace-free Laguerre tensor by L˜ := L− 1n−1 tr(L)g. Inthis paper, we prove a local classification result under the assumption ofparallel Laguerre form and an inequality of the typekL˜k ≤ cR,where c= 1(n−3) √ (n−2)(n−1) is appropriate real constant, depending onthe dimension. 1. IntroductionLet x : M n−1 → R n be an umbilical free hypersurface with non-zero princi-pal curvatures. Let ξ : M → S n−1 be its unit normal. Let {e 1 ,e 2 ,...,e n−1 }be the orthonormal basis for TM with respect to dx · dx, consisting of unitprincipal vectors. Let r i = 1k i , r = r 1 +r 2 +···+r n−1 n−1 be the curvature radius andmean curvature radius of x respectively, where k