Differential Geometry Of Special Null Cratan Curves And Mixed Type Curves In Minkowski 3-Space

In this thesis,we study differential geometry of special null Cartan curves and mixed type curves in Minkowski 3-space.Cylindrical helices are important research objects in differential geometry,and their construction has been thoroughly solved by Izumiya et al in Euclidean 3-space.Because of the different types of tangent vectors of curves,there are non-lightlike general helices and null helices in Minkowski 3-space.The construction of non-lightlike general helices,which is similar to cylindrical helices in Euclidean 3-space,has been solved.However,null helices is fundamentally different from cylindrical helices in Euclidean 3-space.As far as we know,the problem of using a plane curve to construct a null helix has not been solved yet.On the basis of previous work,we give a method that using a special plane curve to construct a null helix,and establish the relationship between plane curves and null helices.We further study the properties of Cartan slant helices,null conical geodesic curves and the null tangential Darboux developable of a null Cartan curve,and we give the relationship between these special curves with the developable surfaces of them.Moreover,we give a classification of singularities of the developable surfaces.In Minkowski 3-space,both non-lightlike curves and null curves are relatively common regular curves.A great deal of research work has been done by predecessors and fruitful results have been obtained.A non-lightlike curve contains only non-lightlike points,and a null curve contains only lightlike points.But a mixed type curve is a kind of regular curve which contains both lightlike points and non-lightlike points,and it is a kind of more general regular curve in Minkowski 3-space.We can use the Frenet frame and the Cartan frame to study non-lightlike curves and null curves respectively.The Frenet frame and the Cartan frame both have no effect on mixed type curves because of the existence of both lightlike points and non-lightlike points.Therefore,the research on mixed type curves was very little and difficult.It was not until 2018 that the study of mixed type curves in the Minkowski plane made any progress.However,the study of mixed type curves is basically blank in Minkowski 3-space,and we even lack the basic tool to study them.As far as we know,no effective research methods have been proposed in this field so far.In order to solve this problem,we construct the lightcone frame in Minkowski 3-space.The lightcone frame provides an effective tool for the study of mixed type curves,and we have been established and proved the basic theorem of mixed type curves by using this frame in Minkowski 3-space.At the same time,as an application of the lightcone frame,we construct the evolute of a mixed type curve in Minkowski 3-space.There are four parts in this thesis.In the chapter 1,we introduce the research background and current situation of singularity theory.Then we introduce the research content and structure of this thesis.In the chapter 2,we mainly introduce the basic concepts and some important conclusions in Minkowski 3-space.Moreover,we introduce the frame of non-lightlike curves and null curves.In the chapter 3,we mainly study special null Cartan curves and developable surfaces.We give a method using a plane curve to construct a null helix,and we also study the properties of Cartan slant helices and null conical geodesic curves.Moveover,we give the relationship between special null Cartan curves and developable surfaces.In the chapter 4,we mainly study mixed type curves in Minkowski 3-space.We give the lightcone frame of Minkowski 3-space,and we establish and prove the existence and uniqueness theorem of mixed type curves by using the lightcone frame in Minkowski 3-space.At the same time,as an application of the lightcone frame,we give the evolutes of mixed type curves.