The Study On Surfaces Immersed In Space Forms

The research presented in this thesis belongs to the theory of submanifolds in differ-ential geometry. Our main content contains:the equiaffine theory for surfaces in affine 4-dimensional space; the theory for slant surfaces in Lorentzian complex space forms;the theory for Lorentzian surfaces in pseudo-Riemannian space forms.In Chapter 3, according to the equiaffine theory for nondegenerate surfaces in R4 estab-lished by Nomizu and Vrancken, we study some classification problems on affine maximal surfaces,locally symmetric surfaces and affine spheres.Under the condition that induced connection is flat,we completely classify affine maximal surfaces in R4.Under the addi-tional condition that affine normal bundle is flat,we completely classify locally symmetric surfaces in R4.In addition,under the condition▽⊥g⊥=0,we characterize all the affine spheres with nonzero constant length mean curvature vector,which indicates that there is great difference between affine umbilical surfaces and affine spheres. Meanwhile, we find some explicit expression of interesting surfaces, which enrich examples of nondegenerate affine surfaces in R4.At last,we classify all the affine translation surfaces with constant Gaussian curvature in R3.In recent years,spatiel surfaces with parallel mean curvature vector in pseudo-Riemannian spaces forms had been completely classified by a series papers of Bang-Yen Chen.Conse-quently, the problem to classify Lorentzian surfaces with parallel mean curvature vector in pseudo-Riemannian space forms becomes interesting and important.Towards this topic, in Chapter 4,we completely classify Lorentzian surfaces with parallel mean curvature vec-tor in pseudo-Euclidean spaces. Furthermore, using isothermal coordinates given by Chen, we classify all the Lorentzian minimal surfaces in a pseudo-Euclidean space,which can be characterized by a family of translation surfaces.In Chapter 5,we mainly study some property of slant surfaces in Lorentzian com-plex space forms.Firstly, we prove a interesting result:Every slant surface in a non-flat Lorentzian complex space form M12(4c) must be Lagrangian.As is well known,there exist many proper slant surfaces in complex projective plane CP2 and complex hyperbolic plane CH2,for instance,see [45,46].Hence,we see that Lorentzian geometry is very different from Riemannian geometry. Moreover,we give a complete classification of pseudo-umbilical slant surfaces in Lorentzian complex space forms.By solving a nonlinear differential equation, we find some interesting examples of slant surfaces.