Rotation Hypersurfaces In Pseudo-Riemannian Space Forms

Rotation hypersurfaces in pseudo-Riemannian space forms are defined and their explicit parametrizations are given in the present paper, and their principal curvatures are computed. In particular, rotation hypersurfaces of finite type are studied and classified. The existence of rotation hypersurfaces with prescribed principal curvature functions in pseudo-Riemannian space forms is proved.This paper is divided into five sections. In section one, the historical background of the relevant problems is presented and the main results are introduced. The definition, the classification, and the explicit parametrizations are given in section two and section three, after that, the principal curvatures are computed. Using the principal curvature formula, we prove an existence theorem of Weingarten hypersurface. In the fourth section, we stress on the study of the finite type rotation hypersurfaces, and obtain the classification of two and three dimensional spherical and hyperbolic rotation surfaces of finite type. It is proved that the parabolic rotation hypersurfaces are minimal or maximal. In section five, the two-ordered ordinary differential equation system that the rotation hypersurface with prescribed principal curvature function satisfies is obtained by using the principal curvature formula and Gauss-Codazzi formula, and the existence of the rotation hypersurfaces of this kind is proved.