四维洛伦兹空间中形状算子不可对角化且有两个不同主曲率类时共形齐性超曲面的分类  

Classification of Time-like Conformally Homogeneous Hypersurfaces with Non-diagonalizable Shape Operator and Two Distinct Curvatures in Four-dimensional Lorentzian Space

作　　者：林燕斌 吕楹 LIN Yanbin;LüYing(College of Mathematics and Statistics,Minnan Normal University,Zhangzhou,Fujian,363000,P.R.China;School of Mathematical Sciences,Xiamen University,Xiamen,Fujian,361005,P.R.China)

机构地区：[1]闽南师范大学数学与统计学院,漳州福建363000 [2]厦门大学数学科学学院,厦门福建361005

出　　处：《数学进展》2024年第3期512-528,共17页Advances in Mathematics（China）

基　　金：福建省自然科学基金资助项目(Nos.2022J05167,2023J01027);福建省中青年教师教育科研项目(No.JAT200319);2020年闽南师范大学校长基金(No.KJ2020002);闽南师范大学校内高级别培育项目(No.MSGJB2023013)。

摘　　要：如果对任意两点p,q∈M_(1)^(3),都存在洛伦兹空间R_(1)^(4)中的一个共形变换σ,使得σ(x(p))=x(q),并且σ(x(M_(1)^(3)))=x(M_(1)^(3)),则称x(M_(1)^(3))为共形齐性超曲面.在本文中我们主要研究形状算子不可对角化且具有2个不同主曲率的类时共形齐性超曲面x:M_(1)^(3)→R_(1)^(4).通过定义共形不变度量gc,典则提升Y,共形切标架{Ei}和典则法标架ξ,我们给出了这类超曲面的一个完备共形不变量系统{E_(1),E_(2),E_(3)}.接下来利用可积性条件构造出了这类超曲面的显式表达式及相应的共形变换子群,从而得到了对这类超曲面的分类定理.A hypersurface x(M_(1)^(3))in Lorentzian space R_(1)^(4)is called conformally homogeneous,if for any two points p,q on M_(1)^(3),there exists a conformal transformationσof R_(1)^(4),such thatσ(x(p))=x(q),σ(x(M_(1)^(3)))=x(M_(1)^(3)).In this paper,we mainly study the time-like conformal hypersurfaces x:M_(1)^(3)→R_(1)^(4)with non-diagonalizable shape operator and two distinct principal curvatures in R_(1)^(4).By introducing the conformal invariant metric g_c,the canonical lift Y,conformal tangent frame Ei and conformal normal frameξ,we find that{E_(1),E_(2),E_(3)}is a complete conformal invariant system for such hypersurfaces.Then we use the integrability conditions to get the explicit expressions of such hypersurfaces and the corresponding conformal transformation subgroups.The corresponding classification theorem is therefore obtained.

关 键 词：不可对角化 共形不变量 共形群 类时共形齐性超曲面 

分 类 号：O186.1[理学—数学]