On The Geometry Of Hypersurfaces In S~2×s~2

In this thesis,we consider the geometry of hypersurfaces in S2 × S2.The main contents of this paper are as follows.In Chapter 1,we introduce the background of this paper and briefly describe the main results of the paper.In Chapter 2,we introduce the basic theory of submanifolds and 3-dimensional locally conformally flat Riemannian manifolds,as well as four classical examples of hypersurfaces in S2 × S2.In Chapter 3,we consider the basic geometric theory of hypersurfaces in S2×S2.Firstly,we obtain the basic geometric theory of hypersurfaces with non-degenerate Jordan angle in S2×S2,including giving structural equations of hypersurfaces with non-degenerate Jordan angle in S2×S2,deriving the second fundamental form of non-degenerate Jordan angle hypersurfaces in S2×S2 is determined by its first fundamental form and Jordan angle function,and constructing the examples of hypersurfaces with non-degenerate and constant Jordan angle in S2×S2.Secondly,we consider the basic geometric theory of hypersurfaces with degenerate Jordan angle in S2×S2.Finally,we obtain the classification theorem for 3-dimensional minimal submanifolds of S5((?))falling in S2×S2.In Chapter 4,we consider locally conformally flat hypersurfaces with constant Jordan angle in S2×S2.On the one hand,we obtain the characterization theorem for locally conformally flat hypersurfaces with non-degenerate and constant Jordan angle in S2×S2.Based on this theorem,we derive that a constant scalar curvature hypersurface with non-degenerate and constant Jordan angle in S2×S2 is a locally conformally flat hypersurface if and only if it is an Einstein hypersurface.On the other hand,we derive that the hypersurfaces with degenerate Jordan angle in S2 ×S2 are locally conformally flat hypersurfaces.As a consequence,we obtain the classification theorem for locally conformally flat hypersurfaces with constant Jordan angle and constant scalar curvature in S2 × S2.In Chapter 5,we consider the hypersurfaces with cohomogeneity one in S2× S2.Firstly,we obtain the basic geometric theory of hypersurfaces with cohomogeneity one in S2×S2.Thus we obtain the characterization theorem of minimal hypersurfaces with cohomogeneity one,and we construct the cohomogeneity one hypersurfaces with arbitrary constant Jordan angle in S2 ×S2.