S_1^(n+1)中Ⅱ型洛伦兹等参超曲面的完全分类  

Complete classification for Lorentzian isoparametric hypersurfaces of type Ⅱ in Lorentzian sphere

作　　者：黎镇琦[1] 姜全德[1] 

机构地区：[1]南昌大学数学系,江西南昌330031

出　　处：《南昌大学学报（理科版）》2012年第3期205-214,共10页Journal of Nanchang University(Natural Science)

基　　金：国家自然科学基金资助项目(10671087);江西省自然科学基金资助项目(2009GZS0017)

摘　　要：研究洛伦兹球面1Sn+1(R1n+2)中的n维Ⅱ型洛伦兹等参超曲面M,给出了这种超曲面的完全分类,证明了这种超曲面的存在性定理和局部刚性定理。如果M的主曲率全都相等,称M是全脐的。设M具有2个互异的主曲率a1,an(a1≠an),形算子A的最小多项式为(λ-a1)2(λ-an)。当a1的重数p=2时,M称为是半脐的。文中证明了M实际上是将乘积流形S+p-1(t)×Sn-p(t)沿着单参数类光直线族{Lt|t∈I}的每一条直线Lt平行移动而得。特别当p=n时是全脐的,当p=2时M是半脐的。In the present paper the Lorentzian isoparametric hypersurfaces M of type Ⅱ in the Lorentzian sphere S1n＋1 are studied. The existence theorem and local rigidity theorem for those hypersurfaces are given, The hypersurface M is totally umbilical if all the principal curvatures of M equal to each other. Suppose M has two distinct principal curvatures al ,an（a1≠an） and the minima polynomial of the shape operator A of M is （λ-al）2 （λ=an）. The hypersurface M is semi-umbilical if the multiplicity of a1 is p=2. tt is proved that M can be obtained by parallel translation of product S＋p-1 （t） × Sn-p （t） of two manifolds along each line Lt in a family { Le It E I} of 1--parameter light-like lines. Particularly M is totally umbilical if p =n, and M is semi-umbilical if p= 2.

关 键 词：洛伦兹球面 洛伦兹超曲面 等参超曲面 

分 类 号：O186[理学—数学]