The Differential Geometry Of Submanifolds In Non-flat Space

In this thesis,we focus on the differential geometry of submanifolds in non-flat space forms.By using the theory of Legendrian or Lagrangian singularities,we solve the classification of singularities of some special submanifolds.In 2003,B.Y.Chen gave the definition of rectifying curve in Euclidean 3-space[23].The particularity of rectifying curve is same as plane curve and spherical curve.We generalize the rectifying curve in non-flat space forms and focus on the geometrical properties of lightlike rectifying curve.Meanwhile,we apply singularity theory to the classification of generic singularities of submanifolds in space with constant scalar curvature.As we know,de Sitter space and anti de Sitter space are the space forms with positive scalar curvature and negative scalar curvature respectively.We stick to the timelike curve in de Sitter space and the spacelike curve in anti de Sitter space,study the geometrical properties of normal hypersurface,focal surface,evolute and one-parameter lightlike hypersurface,point the relationship between the singularities of these submanifolds and some geometrical invariants.At last,we investigate three kinds of pseudo-spherical normal Darboux images of curves on a lightlike surface,classify the generic singularities of pseudo-spherical normal Darboux images,reveal the relationship between the curve and the slice in lightlike surface from the view point of contact geometry,investigate the dual relationship between pseudo-spherical normal Darboux images and Darboux frame fields by using the theory of Legendrian dualities on pseudo-spheres in Minkowski 3-space.There are seven parts in this thesis.In Chapter 1,we review briefly the background of this thesis and introduce the outline of development of this subject in recent years.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 differential geometry related to some non-flat space forms and submanifolds.In Chapter 3,we research the rectifying curves in non-flat space forms and focus on the geometrical properties of lightlike rectifying curve.In Chapter 4,we investigate the classification of the singularities of normal hypersurfaces of de Sitter timelike curves.In Chapter 5,we investigate the geometrical properties of evolutes and focal surfaces of timelike Sabban curves.In Chapter 6,we show the classification of the singularities of one-parameter lightlike hypersurfaces of anti de Sitter spacelike curves.At last,an example is given to explain the main results.In Chapter 7,we discuss three kinds of pseudo-spherical normal Darboux images of curves on a lightlike surface.By using the theory of Legendrian dualities,we investigate the dual relationship between pseudo-spherical normal Darboux images and Darboux frame fields,and an example is also given to explain the main results.