【作者】 王鑫；

【导师】 陈亮；

【作者基本信息】 东北师范大学 ， 基础数学， 2024， 硕士

【摘要】 洛伦兹空间型按常截面曲率可分为三类:截面曲率为正的de Sitter空间,截面曲率为零的Minkowski空间和截面曲率为负的Anti de Sitter空间.本文主要研究三维洛伦兹空间型中混合型贝特朗曲线与特殊曲面的微分几何性质.第一章主要介绍了本文的研究背景及基本知识.第二章首先回顾了三维Minkowski空间中光锥标架曲线的定义并给出了混合型贝特朗曲线的定义及相关的基本性质.其次研究了由光锥标架曲线的法线生成的曲面,称之为法线直纹面,的相关微分几何性质,包括高斯曲率、平均曲率等.进而给出了三维Minkowski空间中混合型贝特朗曲线与光锥标架曲线的法线直纹面的关系.第三章首先给出了三维de Sitter空间中光锥标架曲线的定义及混合型贝特朗曲线定义,进而研究了混合型贝特朗曲线的基本性质.其次研究了三维de Sitter空间中由沿着光锥标架曲线的法线方向的测地线生成的曲面,称之为法测地曲面,的微分几何性质.并研究了混合型贝特朗曲线与光锥标架曲线的法测地曲面之间的关系.第四章首先给出了三维Anti de Sitter空间中光锥标架曲线的定义及混合型贝特朗曲线定义,进而研究了混合型贝特朗曲线的基本性质.其次研究了三维Anti de Sitter空间中由沿着光锥标架曲线的法线方向的测地线生成的曲面,称之为法测地曲面,的相关微分几何性质.最后研究了混合型贝特朗曲线与光锥标架曲线的法测地曲面之间的关系.更多还原

【Abstract】 According to the sectional curvature,the Lorentzian space form is divided into three categories: de Sitter space with positive sectional curvature,Minkowski space with zero sectional curvature and Anti de Sitter space with negative sectional curvature.In this thesis,we mainly study the mixed-type Bertrand curves and special surfaces in Lorentz space from the view point of differential geometry.Chapter 1 introduces the background of the research topic selected in this paper and the basis notions.In Chapter 2,the definion of lightcone framed curve in three dimensional Minkowski space will be firstly introduced,then we give the basis concepts of mixed-type Bertrand curves and study the basic properties of mixed-type Bertrand curves.Moreover,from the view point of differential geometry,we study the properties of the ruled surfaces generated by the normal of the lightcone frame curves.At last,we get the relationship between mixed Bertrand curves and the normal ruled surfaces.In Chapter 3,the 、definion of lightcone framed curve in three dimensional de Sitter space will be firstly introduced,then we give the basis concepts of mixed-type Bertrand curves and we study the basic properties of mixed-type Bertrand curves.Moreover,from the view point of differential geometry,we study the properties of the normal geodesic surfaces which generated by the geodesic in the normal direction of the lightcone frame curves.At last,we get the relationship between mixed Bertrand curves and the normal geodesic surfaces.In Chapter 4,the definion of lightcone framed curve in three dimensional Anti de Sitter space will be firstly introduced,then we give the basis concepts of mixed-type Bertrand curves and we study the basic properties of mixed-type Bertrand curves.Moreover,from the view point of differential geometry,we study the properties of the normal geodesic surfaces which generated by the geodesic in the normal direction of the lightcone frame curves.At last,we get the relationship between mixed Bertrand curves and the normal geodesic surfaces.更多还原