Laguerre Geometry of Hypersurfaces in $\R^n$

Laguerre geometry of surfaces in $\R^3$ 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 $\R^n$. For any umbilical free hypersurface $x: M\to\R^n$ 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 $\R^n$ 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 $\R^n$, the Lorentzian space $\R^n_1$ and the degenerate space $\R^n_0$ we define three Laguerre space forms $U\R^n$, $U\R^n_1$ and $U\R^n_0$ and define the Laguerre embedding $U\R^n_1\to U\R^n$ and $U\R^n_0\to U\R^n$, analogue to the Moebius geometry where we have Moebius space forms $S^n$, \H^n and $\R^n$ (spaces of constant curvature) and conformal embedding \H^n\to S^n and $\R^n\to S^n$ (cf. [4], [10]). Using these Laguerre embedding we can unify the Laguerre geometry of hypersurfaces in $\R^n$, $\R^n_1$ and $\R^n_0$. As an example we show that minimal surfaces in $\R^3_1$ or $\R_0^3$ are Laguerre minimal in $\R^3$.