Some Classification Problems On Submanifolds In Pseudo-Riemannian Space Forms

Pseudo-Riemannian manifold is the differential manifold associated with the pseudo-Riemannian metric. Riemannian manifold belongs to pseudo-Riemannian manifold since the Riemannian metric is the particular example of pseudo-Riemannian metric. Pseudo-Riemannian manifold with constant sectional curvature is called pseudo-Riemannian space form, which is isometric to pseudo-Euclidean space, pseudo-sphere and pseudo-hyperbolic space. Particular, Riemannian space form is isometric to Euclidean space, Euclidean sphere and hyperbolic space. In this paper, We study some classification problems on submanifolds in pseudo-Riemannian space forms according to the invariant of submanifolds.Chen Bang-Yen gave complete classification of spatial surfaces with parallel mean cur-vature vector in pseudo-Euclidean space and obtained 16 families of this kind of surfaces, which contain 3 families in E24. The classification of Lorentz surfaces is more complex than spatial surfaces since the Lorentz metric is indefinite. We characterize all the Lorentz sur-faces with parallel mean curvature vector immersed in pseudo-Euclidean space E24. We conclude that there exist 22 families of this kind of surfaces, which contain 10 families of marginally trapped lorentz surfaces.On the other hand, we consider linear Weingarten spacelike hypersurfaces immersed in de Sitter space with R= aH+b, where R and H are the normalized scalar curvature and the mean curvature. This kind of linear Weingarten hypersurfaces are the generalization of hypersurfaces with constant R and hypersurfaces with R= aH. We introduce an elliptic operator and use the method of moving frame to characterize the linear Weingarten hyper-surfaces according to the sectional curvature or the length of the second fundamental form. We find some conditions for hypersurfaces to be totally umbilical or isoparametric. We also consider the spacelike linear Weingarten submanifolds immersed in Spn+p with R= aH+b and find the conditions for the submanifolds to be totally umbilical or isoparametric by introducing the operator and using the method of moving frame.Moreover, when M is a compact submanifold immersed in a unit sphere with parallel mean curvature vector, we introduce a kind of invariant related to the mean curvature and the second fundamental form, using which we obtain a rigidity theorem of the submanifolds. We also make a detailed comparison with others. At last, we choose a local orthonormal frame field on the minimal surfaces with positive gauss curvature immersed in a unit sphere, under which the corresponding shape operators have simple and convenient form.