Differential Geometry Of Singular Curves

In this thesis,we focus on the differential geometry in singular curves in the Euclidean space.In 2009,Saji,Umehara,Yamada proposed the definition of curvature at cuspidal edges,and characterized the Gaussian curvature at cuspidal edge and swallowtail in the article^[93]published in the journal of Annals of Mathematics.This thesis is a typical working on the differential geometry in the neiborhood of a singular point of submanifolds.Dur-ing this period,many mathematicians devoted themselves to the research of geometrical properties at singular points^{[30,58–60,63–66,79,91–98,100,111–114]}.In these thesis,we take the idea of explore the dimension of the space,we can find the relationship between the singular curves and regular curves.We have resolved the problems of differential geometry of regular curves by the research of the regular curves.For example,the moving frame and curvature and pedal curves of the singular curves and so on.We mainly do research about pedal curves of singular curves in the Euclidean plane and the Euclidean sphere.At the same time,we also do research about evolutes of singular curves in the Minkowski plane.On the other hand,According to the E.Study theory,we investigate a ruled surface as a curve on the dual unit sphere by E.Study's theory.Then we define the notion of evolutes of dual spherical curves for ruled surfaces and establish the relationships between singularities of these subjects and geometric invariants of dual spherical curves.There are five parts in this thesis.In Chapter 1,we introduce the outline of development of this subject in recent years and review briefly the background of this thesis.Moreover,we introduce the structure of the full thesis and describe the main content of this thesis.In Chapter 2,we present the basic notations and results in both differential geometry and singularity theory of pedal curves of the singular curves in the Euclidean space.In Chapter 3,we present the differential geometry of singular curves in the Euclidean sphere and study the properties of pedal curves of singular curves in the Euclidean sphere.In Chapter 4,we investigate a ruled surface as a curve on the dual unit sphere by E.Study's theory.Then we define the notion of evolutes of dual spherical curves for ruled surfaces and establish the relationships between singularities of these subjects and geometric invariants of dual spherical curves.In Chapter 5,we introduce the singular curves and the differential geometry of sub-manifolds of singular curves in the Minkowski plane.