Laguerre Geometry of Hypersurfaces in Rn

Laguerre geometry of surfaces in R is given in the book of Blaschke [1], and have been studied by E.Musso and L.Nicolodi [5], [6], [7], B. Palmer [8] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in Rn. For any umbilical free hypersurface x : M → Rn with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator S : TM → TM , and show that {g, S} is a complete Laguerre invariant system for hypersurfaces in Rn with n ≥ 4. We calculate the EulerLagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space Rn, the Lorentzian space Rn 1 and the degenerate space Rn 0 we define three Laguerre space forms URn, URn 1 and URn 0 and define the Laguerre embedding URn 1 → URn and URn 0 → URn, analogue to the Moebius geometry where we have Moebius space forms Sn, Hn and Rn (spaces of constant curvature) and conformal embedding Hn → Sn and Rn → Sn (cf. [4], [10]). Using these Laguerre embedding we can unify the Laguerre geometry of hypersurfaces in Rn, Rn 1 and Rn 0 . As an example we show that minimal surfaces in R 1 or R 0 are Laguerre minimal in R. 2000 Mathematics Subject Classification: Primary 53A40; Secondary 53B25.