The Lightcone Dualities for Submanifolds in the Sphere [an abstract of dissertation and a summary of dissertation review]

In this paper we consider the submanifolds in the lightcone unit sphere. The unit sphere can be canonically embedded in the lightcone and de Sitter (n+1)-space in the Lorentz-Minkowski space. We investigate these submanifolds in the framework of the theory of Legendrian dualities between pseudo-spheres in the Lorentz-Minkowski space. We compare these dualities with the classical duality of submanifolds in the Euclidean sphere.