Laguerre Geometry of Hypersurfaces in ${\mathbb{R}}^n$

Abstract

Laguerre geometry of surfaces in ${\mathbb{R}}^3$ 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 ${\mathbb{R}}^n$. For any umbilical free hypersurface $x:M\to {\mathbb{R}}^n$ with non-zero principal curvatures we define a Laguerre invariant metric $g$ on $M$ and a Laguerre invariant self-adjoint operator $\mathbb{S}:TM\to TM$, and show that $\{g,\mathbb{S}\}$ is a complete Laguerre invariant system for hypersurfaces in ${\mathbb{R}}^n$ with $n\ge 4$. 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 Lorentzian space ${\mathbb{R}}_1^n$ and the degenerate space ${\mathbb{R}}_0^n$ we define three Laguerre space forms $U{\mathbb{R}}^n$, $U{\mathbb{R}}_1^n$ and $U{\mathbb{R}}_0^n$ and define the Laguerre embedding $U{\mathbb{R}}_1^n\to U{\mathbb{R}}^n$ and $U{\mathbb{R}}_0^n\to U{\mathbb{R}}^n$, analogue to the Moebius geometry where we have Moebius space forms $S^n$, $\H^n$ and ${\mathbb{R}}^n$ (spaces of constant curvature) and conformal embedding $\H^n\to S^n$ and ${\mathbb{R}}^n\to S^n$ (cf. [4], [10]). Using these Laguerre embedding we can unify the Laguerre geometry of hypersurfaces in ${\mathbb{R}}^n$, ${\mathbb{R}}_1^n$ and ${\mathbb{R}}_0^n$. As an example we show that minimal surfaces in ${\mathbb{R}}_1^3$ or ${\mathbb{R}}_0^3$ are Laguerre minimal in ${\mathbb{R}}^3$.

24 pages

Cited by in corpus (1)

• Möbius and Laguerre geometry of Dupin Hypersurfaces