Laguerre Hypersurfaces with definite Laguerre tensor

Abstract

$x:M^{n-1}\rightarrow \mathbb{R}^{n}$ be an umbilical free
hypersurface with non-zero principal curvatures.
Then $x$ is associated with a Laguerre
metric $\mathbf{g}$, a Laguerre tensor $\mathbf{L}$, a Laguerre form
$\mathbf{C}$, and a Laguerre second fundamental form $\mathbf{B}$,
which are invariants of $x$ under Laguerre transformation group.
In this paper, we will study the Laguerre hypersurface with
parallel Laguerre form and nonnegative or non-positive Laguerre tensor.