On Differential Geometry Of Submanifolds In Minkowski 4-space

This paper is to study the di?erential geometry of spacelike curves, timelike surfacesand timelike hypersurfaces in Minkowski 4-space by singularity theory and Lorentziangeometry. It splits naturally into three parts.Part I is to study spacelike curves in Minkowski 4-space. The spacelike curve inMinkowski 4-space is di?erent from curve in Euclidean space and spacelike curve inMinkowski 3-space. For the spacelike curve, there is a 2-dimensional lightcone in nor-mal space, so there is a lightcone Gauss image family. We can study the lightcone Gaussimages and the spacelike curves by the study on the lightcone Gauss image family. Inorder to study the spacelike curves, we ?rst give local di?erential geometry of the spacelikecurves. After that, we construct lightcone height functions, which are useful tools for thestudy of singularities of the lightcone Gauss images, and then show the relations betweensingularities of the lightcone Gauss images and that of the lightcone height functions. Atlast, we give classi?cation of singularities of the lightcone Gauss images.Part II is to study timelike surfaces in Minkowski 4-space. As it was expected,the situation gives certain peculiarities when it is compared with spacelike surface inMinkowski 3-space and surface in Euclidean space. For instance, in our case it is alwayspossible to choose two lightlike tangent directions along the surface as a frame of itstangent bundle. By using this, we de?ne a Lorentzian invariant and call it the tangentlightcone curvature of the timelike surface. The ?ash point of the timelike surface is thatthere is a lightcone in its tangent space, so we decide to study this surface from tangentspace. In order to do that, we ?rst give local di?erential geometry of the timelike surfacefrom tangent space viewpoint. Then we study singularities of tangent lightcone mapsby Legendrian singularity theory and Montaldi's contact theory. After these, we studygeometric properties of these singularities.Part III is to study timelike hypersurfaces in Minkowski 4-space. There is a 2- dimensional lightcone in tangent space of the timelike hypersurface, so there are a lot oflightlike directions. In order to study the relation between de Sitter Gauss curvature andlightlike curvature, we review the Lagrangian singularity theory, and discuss singularitiesof the de Sitter Gauss map by the contact between timelike hypersurface and spacelikehyperplane.