On Laguerre Form and Laguerre Isoparametric Hypersurfaces

Let x ： M^n-1→ R^nbe an umbilical free hypersurface with non-zero principal curvatures.M is called Laguerre isoparametric if it satisfies two conditions, namely, it has vanishing Laguerre form and has constant Lauerre principal curvatures. In this paper, under the condition of having constant Laguerre principal curvatures, we show that the hypersurface is of vanishing Laguerre form if and only if its Laguerre form is parallel with respect to the Levi-Civita connection of its Laguerre metric.

R^n中具有3个不同主曲率的Laguerre等参超曲面

若超曲面的Laguerre形式为零且Laguerre第二基本形式的特征值(称为Laguerre主曲率)为常数,则称超曲面为Laguerre等参超曲面.对Rn中具有3个不同主曲率的Laguerre等参超曲面进行了研究,得到了相应的分类定理.

Laguerre Isopararmetric Hypersurfaces in R^4

Let x ： M →R^n be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and a Laguerre second fundamental form B which are invariants of x under Laguerre transformation group. A hypersurface x is called Laguerre isoparametric if its Laguerre form vanishes and the eigenvalues of B are constant. In this paper, we classify all Laguerre isoparametric hypersurfaces in R^4.

Complete Hypersurfaces with Constant Laguerre Scalar Curvature in R^n

Let x ： M^n-1→ R^n be an umbilical free hypersurface with non-zero principal curvatures. Two basic invariants of M under the Laguerre transformation group of Rn are Laguerre form C and Laguerre tensor L. In this paper, we prove the following theorem： Let M be an （n-1）-dimensional （n 〉 3） complete hypersurface with vanishing Laguerre form and with constant Laguerre scalar curvature R in R^n, denote the trace-free Laguerre tensor by L = L -1/n-1tr（L）· Id. If supM ｜｜L｜｜=0,then M is Laguerre equivalent to a Laguerre isotropic hypersurface; and if supM ｜｜L^-｜｜≡√（n-1）（n-2）R/ （n-1）（n-2）（n-3） , M isLaguerre equivalent to the hypersurface x^- ： H^1× S^n-2 → R^n.

R^5中的Laguerre等参超曲面(英文)

设M是R^n中不含脐点的超曲面,在M上可以定义所谓的Laguerre度量g,Laguerre张量L和Laguerre形式C,Laguerre第二基本形式B,它们都是M在Laguerre变换群下的不变量.由Laguerre几何的经典定理可知,在Laguerre等价下M(n>3)完全由Laguerre度量g和Laguerre第二基本形式B确定.如果M满足Laguerre形式恒等于零而且Laguerre第二基本形式关于Laguerre度量的特征值均为常数,则称之为Laguerre等参超曲面.易知,所有Laguerre等参超曲面都是Dupin超曲面.本文在[Acta Math.Sin.,Engl.Ser.,2012,28(6):1179-1186]的基础之上研究R^5中的Laguerre等参超曲面,得到了分类定理.

R^n上具有三个主曲率的Laguerre等参超曲面

R^n上主曲率非零的定向无脐超曲面x:M→R^n称为Laguerre等参超曲面,如果它的Laguerre形式C=∑_iC_iω_i=∑_iρ^(-1)(E_i(logρ)(r-r_i)-E_i(r))ω_i为零,Laguerre形状算子S=ρ^(-1)(S-rid)的特征值为常数,这里ρ~2=∑_i(r-r_i)~2,r=r=((r_1)+r_2+…+r_(n-1))/(n-1)是平均曲率半径,S是x的形状算子,{E_i}是Laguerre度量g的单位正交标架,{ω_i}是对偶标架.本文给出R^n上具有三个互异Laguerre主曲率的Laguerre等参超曲面的分类.