Differential Geometry Of Framed Curves And Related Surfaces In Three-Dimensional Space

In this thesis,we mainly investigate the differential geometry of framed curves and related surfaces in three-dimensional space.The classification and recognition of generic singularities of framed curves and related surfaces which allow singularities in Euclidean3-space and Minkowski 3-space are emphatically discussed.In the study of differential geometry,Frenet frame is a classical tool for studying regular curves in Euclidean 3-space.We can obtain most information of a curve by using the Frenet frame.However,if the curve contains the points at which the curvature of the curve vanishes,then the traditional Frenet frame will fail at these points.In order to solve such problems,R.Bishop proposed the concept of Bishop frame for regular curves in Euclidean 3-space in1975.He pointed out that the Bishop frame still exists at the point of vanishing curvature.However,the results related to Bishop frame have not involved the study of degenerate curves up to now.Therefore,in this thesis,we define(1,k)-type Bishop framed curve in Euclidean 3-space,which is an extension of the traditional Frenet curve,and also realizes the original intention of R.Bishop when he proposed the Bishop frame.Inspired by R.Bishop,S.Honda and M.Takahashi proposed the concept of the framed curve in Euclidean space in 2016.However,the emphases of them are different.The Bishop frame focuses on the study of high-order degenerate regular curves,while the framed curve pays more attention to the study of first-order degenerate singular curves.In particular,the framed curve also plays an important role in the construction of singular submanifolds.In this thesis,we introduce the framed curve into Minkowski 3-space and give the definition of the spacelike framed curve in Minkowski 3-space.Furthermore,spacelike curves with singularities and related geometric objects can be studied.The main research contents and results of this thesis are as follows:Firstly,we give the definition of the adjoint curve of a Bishop curve in Euclidean 3-space.Then,we show the geometric meaning of Bishop curvatures by the adjoint curve of a Bishop curve.Furthermore,we study the evolute and focal surface of a(1,k)-type Bishop framed curve in Euclidean 3-space,and classify the singularities of the focal surface.Secondly,we introduce the concept of spacelike framed curves with non-lightlike components into Minkowski 3-space.Then,we establish the local fundamental theory,and prove the existence and uniqueness theorems of such spacelike framed curves.As the applications of this theory,we study the lightcone Gauss map,lightcone pedal curve and lightcone developable surface of this kind of spacelike framed curves,and obtain the equivalence conditions that they are diffeomorphism to the standard singularity models at their singularities,and give the corresponding recognition theorems for singularities.Finally,since the most essential difference between Minkowski space and Euclidean space is the existence of lightlike vectors.Moreover,considering the characteristics of a spacelike curve itself,we introduce the concept of spacelike framed curves with lightlike components in Minkowski 3-space.Then,we study the evolute and focal surface of a spacelike framed curve with lightlike components.We prove that the cuspidal edge and swallowtail singularities of the focal surface are stable in generic,and give the classification and recognition theorems of the generic singularities of the focal surface.