Global properties of codimension two spacelike submanifolds in Minkowski space

Abstract

Abstract. We consider codimension two spacelike submanifolds with a parallel normal field (i.e. vanishing normal curvature) in Minkowski space. We use the analysis of their contacts with hyperplanes and hyperquadrics in order to get some global information on them. As a consequence we obtain new versions of Carathéodory's and Loewner's conjectures on spacelike surfaces in 4-dimensional Minkowski space and 4-flattenings theorems for closed spacelike curves in 3-dimensional Minkowski space.

Spacelike surfaces in anti de Sitter four-space from a contact viewpoint

“…Kronecker curvature of M is defined as K We say that the embedding X is generic if its associated horospherical height functions family is structurally stable (see [14]), in other words, if X is a generic embedding in H On the other hand, as seen in [12], when M is a closed orientable surface in H 3 + (−1), we can consider a globally defined hyperbolic Gauss map on M (and thus on M ), and consequently, a globally defined Gauss-Kronecker curvature function on M (and thus on M ). For the purposes of the following result, we can either fix the superindex + or − in the above arguments and denote by K hM the globally defined Gauss-Kronecker curvature function on M and by K h the globally defined Gauss-Kronecker curvature function on M .…”

“…It is well-known that the Lorentzian space form with zero curvature is Lorentz-Minkowski space and with positive curvature is de Sitter space. These Lorentzian space forms have been well studied (cf., [10,11,13,14,15,16]). However, there are not much results on submanifolds immersed in Anti de Sitter space, in particular from the viewpoint of singularity theory.…”

“…In the study of submanifolds in Lorentzian-Minkowski space, the codimension two spacelike submanifolds happen to be the most interesting subjects, both from the view point of singularity theory and of the theory of relativity (cf., [10,11,13,14,15,16]). For instance, one of the important objects in the theory of relativity are the lightlike hypersurfaces because they provide good models for different types of horizons [6,20].…”

“…This Gauss map has an associated horospherical Gauss-Kronecker curvature as well as principal configurations. We state some global properties concerning these, obtained by pull-back of the corresponding results on surfaces in H 3 + (−1) ( [12,14]). …”

Lightlike hypersurfaces along spacelike submanifolds in anti-de Sitter space

“…Anti-de Sitter space is a Lorentzian space form with negative curvature. Recently, submanifolds in Lorentz-Minkowski space or de Sitter space have been well investigated (cf., [8,11,13,14,19,20]). However, there are not so many results on submanifolds in anti-de Sitter space, in particular from the viewpoint of singularity theory.…”