Legendrian Dualities Of Submanifolds And Classification Of Their Singularities

In this thesis, taking Legendrian dualities as the main line, we solve the classi?cation of singularities of some submainfolds which can be seen as Legendrian dualities by using the theory of Legendrian or Lagrangian singularities. We not only ?nd a kind of Legendrian dualities theorem for one parameter pseudo-spheres families in pseudo Euclidean space, but also study the extrinsic geometry of Lorentzian hypersurfaces in Anti de sitter space, psedou sphere or nullcone respectively by using this theorem. We prove that there are nullcone Gauss maps and φ-pseudo-sphere Gauss maps of these Lorentzian hypersurfaces by using the theorem of Legendrian dualities. Secondly, we apply the theory of Lagrangian singularities to the classi?cation of the generic singularities of anti de Sitter Gauss maps of the Lorentzian hypersurfaces in R42. Additionally, we classify the generic singularities of dual surfaces of spacelike curves in de Sitter space in the frame work of the theory of Legendrian dualities between pseudo-spheres in Minkowski space. Finally, we investigate the dual relationship between spherical indicatrixes of regular curves by using the theory of Legendrian dualities on spheres in Euclidean 3-space; Furthermore, we classify the generic singularities of Bishop spherical indicatrixes, Bishop spherical Darboux image, Bishop dual surfaces and Bishop ruled surfaces of regular curves in the frame work of relatively adopted parallel frame and the theory of Lagrangian singularities.The thesis is organized as follows.In Chapter 1, we ?rst introduce the outline of development of this subject in recent years and review brie?y the background of the subject of this thesis. Moreover, we introduce the structure of the full thesis and describe the main content of this thesis.In Chapter 2, we introduce some basic notions and results of the theory of Legendrian singularities and Lagrangian singularities.In Chapter 3, we prove the theorem of Legendrian dualities for one parameter pseudospheres families in pseudo Euclidean space and apply it to the study of the geometry of Lorentzian hypersurfaces in Anti de sitter space, psedou sphere Sn2 or nullcone respectively.For Lorentzian hypersurfaces in R42, we classify the generic singularities of their Anti de Sitter Gauss mappings by using the theory of Lagrangian singularities.In Chapter 4, we investigate generic singularities of dual surfaces of spacelike curves in de Sitter space. There are two pairs of spacelike dual surfaces of spacelike curves, which are called the ?rst lightcone dual surfaces, the second lightcone dual surfaces, the ?rst hyperbolic dual surfaces and the second hyperbolic dual surfaces according to the space in which its position vector is located. We prove that the ?rst lightcone dual surface has generic singular points, such as cuspidal edge and swallowtail, but the second lightcone dual surface is regular. So does the hyperbolic dual surfaces. With the help of Legendrian dualities, we reveal the relationships among the lightcone dual surfaces, the hyperbolic dual surfaces and the spacelike curves. We ?nd two invariants which can be used to describe these singularities, by investigating the properties of the lightcone height functions and timelike height functions. We will interpret the applications and illustrations of the main results by taking an example.In Chapter 5, we investigate the dual relationship between spherical indicatrixes of regular curves by using the theory of Legendrian dualities on spheres in Euclidean 3-space;Additionally, we classify the generic singularities of Bishop spherical indicatrixes, Bishop spherical Darboux images, Bishop dual surfaces and Bishop ruled surfaces of regular curves in the frame work of the relatively adopted parallel frame and the theory of Lagrangian singularities. We also get some properties of Bishop slant helix. Finally, two examples are given to explain the main results.