Null Surfaces Of Null Cartan Curves And Evolutes Of Framed Curve In 3-dimensional Space

In this thesis,we mainly consider the differential geometry of null Cartan curves in H₁³ which are regarded as the submanifolds in 4 dimensional semi-Euclidean space with index 2.Firstly,According to the characteristics of the Cartan curve,we establish the Frenet frame and get the special Cartan Frenet equation.Then we define the ruled null surface and the binormal indicatrix height functions associated with null cartan curves.we investigate the relationships between the geometric invariants of null Cartan curves and the singularities of the ruled null surfaces under the actions of Lorenzian group.and classify the singularities of the ruled surfaces by use of singularity theory.In addition,this thesis gives the definition of the evolutes of framed curves in the 3-dimensional Euclidean space and studies the properties of evolutes.We remark that this new definition on the evolute is consistent with the classical one when the curve is a regular curve.There are four parts in this thesis.In Chapter 1,we mainly introduces the development of singularity theory,and briefly describes the researches and the basic framework of this thesis.In Chapter 2,we present the preliminary knowledge for the semi Euclidean space R₂⁴with index 2.we introduce the basic concept such as the pseudo inner product,the pseudo vector product,pseudo orthogonal,pseudo arc length parameter,spacelike vector,null vector,and timelike vector spacelike curve,Lightline curve?null curve?and timelike curve etc.,and gives the concept of some important submanifolds like three dimensional Anti-de Sitter space,three dimensional unit pseudo sphere with index 2 and three dimensional open light cone etc..In particular,we give a conclusion[29]which can guarantee the unique null vector matching with the null tangent vector.In Chapter 3,we Mainly introduces the the Frenet frame and Frenet equation of null Cartan curve of three dimensional Anti-de Sitter space.As a basic research tool and using the singularity classification method of Bruce,we classify the singularities of the ruled surfaces of which the base curve is the principal normal indicatrix of null Cartan curve.The specific classification results are shown in theorem 3.6.1.In Chapter 4,we study the properties of evolute of curves with singularities.we firstly give the concept of framed curve and define a moving adapted frame for the framed curve.By using the moving adapted frame,we define some smooth functions like as the curvature of a regular curve.These functions are called framed curvature.On this basis,we give the definition of the evolutes of framed curves in the 3-dimensional Euclidean space.For the specific definition,see theorem 4.2.1.We remark that this new definition on the evolute is consistent with the classical one when the curve is a regular curve.