Laguerre geometry of hypersurfaces in ℝⁿ

Laguerre geometry of surfaces in ℝ³ is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321-4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123-138 (1999); Int. J. Math. 11(7), 911-924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281-293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in ℝⁿ. For any umbilical free hypersurface x: M → ℝ^n/ with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator double script S sign : TM → TM, and show that g doubl script S sign is a complete Laguerre invariant system for hypersurfaces in ℝⁿ with n ≥ 4. We calculate the Euler-Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space ℝⁿ, the semi-Euclidean space ℝ₁ⁿ and the degenerate space ℝ₀ ⁿ we define three Laguerre space forms Uℝⁿ , Uℝ₁ⁿ and ℝ₀ⁿn and define the Laguerre embeddings Uℝ₁ⁿ → U ℝⁿ and ℝ₁ⁿ, analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , ℍⁿ and ℝⁿ (spaces of constant curvature) and conformal embeddings ℍⁿ → Sⁿ and ℝⁿ → Sⁿ [cf. Liu et al. in Tohoku Math. J. 53, 553-569 (2001) and Wang in Manuscr. Math. 96, 517-534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in ℝⁿ, ℝ₁ⁿ and ℝ₀ ⁿ. As an example we show that minimal surfaces in ℝ₁³ or ℝ₀³ are Laguerre minimal in ℝ³.