Dualities Of The Submanifold In The Null Sphere

In this thesis,we mainly investigate the differential geometry of the curves and framed surfaces with singular points on the null de Sitter sphere.The submanifolds with singular points which located on the null de Sitter sphere are major research objects in differential geometry,the previous research is only focused on the regular condition,it is of great significance to investigate the singular curves and framed surfaces on the null sphere in semi-Euclidean space.The essential difference between the semi-Euclidean space and the Euclidean space is the existence of light vector.Since they are degenerate,so we would like to solve these problems by using the duality theory and the results of our previous studies in Euclidean space.Thus we adopt its dual curve and dual surface by using homeomorphic theory in the Euclidean space,so that they have the same properties as the original curves.Furthermore,we extended our investigation of the regular spacelike curves to the case where the singularities are allowed.On the one hand,we define the nullcone evolute and nullcone involute of the singular spacelike curve,as well as discuss the relationships among them and give their singularity types.On the other hand,as a generalization,we construct the nullcone dual worldsheet generated by singular spacelike curve.The worldsheet,being defined as a two-dimensional manifold generated by a string which moves through spacetime.When the set of parallel light rays travels in the time direction(i.e.normal direction),its traveling trajectories can be regarded as the timelike surface which is called worldsheet.As an important geometry object,some worldsheets will inevitably have the singularities of geometry under the effects of gravity.We characterize the singularity type of the nullcone dual worldsheet by means of the spherical geometric invariant.Then we also establish the special dual relationship between singular spacelike curve and worldsheet.The structure of this thesis is established as follows.First,from the viewpoint of Legendrian singularity theory,we study the future nullcone dual hypersurfaces and the future nullcone surfaces of regular spacelike curve on the null de Sitter sphere.We obtain the projections of these two critical value sets on the de Sitter sphere are same,and it is exactly to be the spherical evolute of the original curve.Moreover,we also classify the singularities of the future nullcone dual hypersurfaces,the future nullcone surfaces and spherical evolute are diffeomorphic to different types of singularities under appropriate conditions.Second,we extended our investigation of the regular spaceike curves to the case where the singularities are allowed.We investigate the nullcone dual surfaces of singular spacelike curves in the null sphere,we discuss the dual relationship between them.Then we focus our attention on the study of the relationships among them and give their singularity types.As an application,we investigate the nullcone dual worldsheets of singular spacelike curves,we characterize the types of singularities of the nullcone dual worldsheets by the new spherical invariants.Futhermore,we demonstrate that there exists a special dual relationship between the nullcone dual worldsheets and singular spacelike curves.Finally,we investigate the framed surface and its dual surface which may have sigularities on the null de Sitter sphere.It is of great significance to explore the classifications of singularities of these framed surfaces.Meanwhile,we further show that the framed surfaces as one-parameter families of framed curves and establish the relations between them.