16 - Spacelike submanifolds of codimension at most two in de Sitter space

Summary

Abstract

The aim of this paper is to provide a description of the main geometrical properties of spacelike submanifolds of codimension at most two in de Sitter space, that have been studied by the author with full details in other papers, as an application of the theory of Legendrian singularities. We analyze the geometrical meaning of the singularities of lightcone Gauss images, lightcone Gauss maps and lightlike hypersurfaces of generic spacelike surfaces in de Sitter 3-space and de Sitter 4-space.

Introduction

In this paper we consider de Sitter space, which is a Lorentzian space form with positive curvature defined by a pseudo n-sphere in Minkowski space. The spacelike curves in de Sitter 3-space are investigated in [5] and the lightlike surface of the spacelike curves are constructed from the Frenet-Serret type formula. In [9] the differential geometry of the timelike surfaces in de Sitter space are discussed, and the singularities of de Sitter Gauss images of timelike surfaces in de Sitter 3-space are classified. The principal, asymptotic and characteristic curves associated to the de Sitter Gauss maps are investigated in [10], and the contact of timelike surfaces with geodesic loci are investigated in [11]. In [12] we investigated the lightcone Gauss image of spacelike hypersurface in de Sitter space, which is the analogous tool in [6]. The singularities of the Gauss images correspond to the parabolic sets of spacelike hypersurfaces.