Laguerre geometry of hypersurfaces in $\mathbb{R}^{n}$

Abstract

Laguerre geometry of surfaces in $\mathbb{R}^{3}$ 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 $\mathbb{R}^{n}$. For any umbilical free hypersurface $x: M \rightarrow \mathbb{R}^{n}$ with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator $\mathbb{S}$: TM → TM, and show that $\{g, \mathbb{S}\}$ is a complete Laguerre invariant system for hypersurfaces in $\mathbb{R}^{n}$ 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 $\mathbb{R}^{n}$, the semi-Euclidean space $\mathbb{R}^{n}_{1}$ and the degenerate space $\mathbb{R}^{n}_{0}$ we define three Laguerre space forms $U\mathbb{R}^{n}$, $U\mathbb{R}^{n}_{1}$ and $U\mathbb{R}^{n}_{0}$ and define the Laguerre embeddings $U\mathbb{R}^{n}_{1} \rightarrow U \mathbb{R}^{n}$ and $U\mathbb{R}^{n}_{0} \rightarrow U \mathbb{R}^{n}$, analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ, $\mathbb{H}^{n}$ and $\mathbb{R}^n$ (spaces of constant curvature) and conformal embeddings $\mathbb{H}^n \rightarrow S^n$ and $\mathbb{R}^n \rightarrow S^n$ [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 $\mathbb{R}^n$, $\mathbb{R}^n_1$ and $\mathbb{R}^n_0$. As an example we show that minimal surfaces in $\mathbb{R}^3_1$ or $\mathbb{R}_0^3$ are Laguerre minimal in $\mathbb{R}^3$.