The Special Curves And Surfaces In 3 Dimensional Space Forms

In this thesis,we pay attention to the differential geometry of special curves and special surfaces in 3-space forms.In 2002,Izumiya and Takeuchi studied the relationship between Bertrand curves and ruled surfaces from the view point of geometry of curves on ruled surfaces in 3 dimensional Euclidean space.Motivated by their ideas,we discuss the connection between Bertrand curves and geodesic surfaces in 3-space forms.For a singular curve in Euclidean space,using the idea of uplifting the dimension of the submanifold,we can construct the moving frame of a singular curve and we can investigate the geometrical properties of the curve at the singular points.We apply the above method to discuss the geometrical properties of singular Bertrand curves and singular Mannheim curves in 3-space forms.At last,we focus on the non-developable ruled surfaces along the singular curve in 3 dimensional Euclidean space.The non-developable ruled surfaces are principal normal surfaces and binormal surfaces.We reveal the relationship between the singular points of a curve and the singular points of a surface,give the classification of generic singularities of nondevelopable ruled surfaces from singularity theory,and describe the special non-developable ruled surfaces by using the structure functions.There are five parts in this thesis.In chapter 1,we review the background of singular theory and the development of this subject in recent years.Moreover,we introduce the main content of this thesis and give the structure of the full thesis.In chapter 2,we present the basic definition in the semi-Euclidean space related to some non-flat space form and submanifolds.In chapter 3,we study the geometrical properties of Bertrand curves in non-flat 3-dimensional space forms and from the view point of geometry of curves on ruled surfaces to discuss the connection between Bertrand curves and geodesic surfaces.In chapter 4,we consider the geometrical properties of singular Bertrand curves and singular Mannheim curves in the 3 dimensional Riemannian space forms and give some examples.In chapter 5,we show the classification of the non-developable ruled surfaces of singular curves in the 3 dimensional Euclidean space and give concrete examples.