The Lightcone Dualities For The Submanifold In The Sphere

In this thesis, we studied the submanifolds in the lightcone unit sphere in the frame-work of the theory of Legendrian dualities between pseudo-spheres in Minkowski space.The subjects we studied include the curves in the unit2-sphere, the curves in the unit3-sphere and the hypersurfaces in the unit n-sphere. The lightcone unit sphere can benaturally embedded in the lightcone and the de Sitter space. We consider the Legendriandualities between lightcone and lightcone and the Legendrian dualities between lightconeand de Sitter space. Then we got the lightcone dual hypersurface of the submanifold inthe lightcone unit sphere and the lightcone dual hypersurface of the submanifold in theEuclidean unit sphere. We studied the singularities of these hypersurfaces. When we com-pared these singularities, we got some interesting results. We also gave the classifcationsof the singularities of these hypersurfaces.The thesis is organized as follows.In Chapter1, we frst reviewed briefy the background of the subject of lightconeunit sphere and introduced the outline of development of this subject in recent years.Moreover, we introduced the structure of the full thesis and described the main contentof this thesis.In Chapter2, we introduced the basic preliminary knowledge in this thesis.In Chapter3, we mainly introduced the curves in the unit2-sphere. We gave theFrenet-Serret type formulae for the curve in the unit2-sphere. According to the formulae,we gave the lightcone dual hypersurface of the curve in the lightcone unit2-sphere andthe lightcone dual hypersurfaces of the curve in the Euclidean unit2-sphere. We gave theclassifcations of the singularities of these two lightcone dual hypersurfaces.In Chapter4, we mainly studied the curves in the unit3-sphere. We gave the Frenet-Serret type formulae for the curve in the unit3-sphere. According to the formulae, wegave the lightcone dual hypersurfaces of the curve in the lightcone unit3-sphere and the lightcone dual hypersurfaces of the curve in the Euclidean unit3-sphere. As the mainresult, we gave the classifcations of the singularities of these lightcone dual hypersurfaces.In Chapter5, we mainly studied the hypersurfaces in the unit n-sphere. We gavethe lightcone dual hypersurface of the hypersurface in the lightcone unit n-sphere andthe lightcone dual hypersurface of the hypersurface in the Euclidean unit n-sphere. Wecalculated the critical set of these hypersurfaces and we defned them as the lightcone focalsurface of the hypersurfaces respectively. We also studied the contact of hypersurfaceswith the parabolic (n1)-spheres and parabolic n-hyperquadric as an application of thetheory of Legendrian singularities.