Singularities Of De Sitter Surfaces And Gauss-Bonnet Theorem In Translational Rotation Groups

This thesis mainly studies the singularity classification problem of de Sitter surfaces generated by timelike curves on timelike hypersurfaces in four-dimensional Minkowski space-time and the Gauss-Bonnet theorem on translational rotation groups.Firstly,by characterizing the singularity of the spacelike tangential height function defined on the timelike curve in a timelike hypersurface in a four-dimensional Minkowski space,the geometric invariant of the singularity type of the de Sitter surface is found.It is further found that the singularity types of de Sitter surfaces are cuspidal edge,swallowtail,and cuspidal beaks,and there is no cuspidal lips type singularity.Secondly,the Gauss-Bonnet theorem on the universal covering group of Euclidea motion group with the general left invariant Riemannian metrics is studied by using the Riemannian approximation.By defining the curvature of the curves,the geodesic curvature and the Riemannian Gaussian curvature of the surfaces in the group,and their corresponding sub-Riemannian limits,their specific expressions are calculated and obtained.Finally,these results are used to prove the Gauss-Bonnet theorem on the translation rotation group.