Laguerre characterization of some hypersurfaces
Bull. Korean Math. Soc. 2016 Vol. 53, No. 3, 875-884
https://doi.org/10.4134/BKMS.b150386
Published online May 31, 2016
Jianbo Fang and Fengjiang Li
Chuxiong Normal University, Yunnan Normal University
Abstract : Let $x$ : ${M}^{n-1}\rightarrow{\mathbb R}^{n}$ $(n\geq4)$ 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. We denote the Laguerre scalar curvature by $R$ and the trace-free Laguerre tensor by $\tilde{\mathbf{L}}:={\mathbf{L}}-\frac{1}{n-1}tr({\mathbf{L}})\mathbf{g}$. In this paper, we prove a local classification result under the assumption of parallel Laguerre form and an inequality of the type $$\|\tilde{\mathbf{L}}\|\leq cR,$$ where $c=\frac{1}{(n-3)\sqrt{(n-2)(n-1)}}$ is appropriate real constant, depending on the dimension.
Keywords : Laguerre geometry, hypersurfaces
MSC numbers : Primary 53A40, 53B25
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