仿Laguerre特征值及Laguerre等参超曲面研究

设x:M→R~n是主曲率非零的无脐点超曲面,若x满足Laguerre形式C等于0,且所有Laguerre主曲率都是常数,则称x是Laguerre等参超曲面.证明了当Laguerre形式C等于0时,若Laguerre主曲率是常数,则仿Laguerre特征值为常数;若仿Laguerre特征值为常数且λ非零,则Laguerre主曲率是常数.

Para-Laguerre Eigenvalues and Laguerre Isoparametric Hypersurfaces

Let x:M→R~n be an umbilical free hypersurface with non-zero principal curvature,then x is called Laguerre isoparametric hypersurface if it satisfies two conditions that Laguerre form vanishes and all Laguerre principal curvatures are constant.In this paper,under the condition of having vanishing Laguerre form,we show that if the hypersurface is of constant Laguerre principal curvatures,then its para-Laguerre eigenvalues are constant;and if the para-Laguerre eigenvalues are constant andλis not zero,then its Laguerre principal curvatures are constant.