Hypersurfaces With Constant Sectional Curvature Of S~n×S And H~n×S

In this paper, we mainly study hypersurfaces with constant sectional curvature in the product manifolds Sn×S and Hn×S. This paper is divided into four chapters, which are organized as follows:In the first chapter, we firstly introduce the background of our investigation; Secondly, we give some relevent knowledge about the direct product of Riemannian manifolds; Finally, we put forward the questions considered in this paper.In the second chapter, we discuss the complete surface of rotation in S2×S and H2×S. Using the idea of the Codazzi pair and its relevant holomorphic forms, we have the following conclusion:Given a real constant K>(1+ε)/2, Then there exist, up to isometries, a unique complete surface of rotation in S2×S with constant Gaussian curvature K(ε=1), which is given by Theorem 2.2.3, and a unique complete surface of rotation in H2×S with constant Gaussian curvature K(ε=-1), which is given by Theorem 2.2.1. For the completeness of surfaces, we have another important result:There is no complete surface in Qε2×S with constant Gaussian curvature K<-1.In the third chapter, we focus on the constant angle surfaces in S2×S and H2×S. By this we mean a surface that the unit normal makes a constant angle with the tangent to S. Using the integrability theory, the main classification theorem of this chapter can be stated as:Theorem 3.3.1 An immersionφ:M2→Qε2×S is a surface of constant angleθ∈[0,π/2] if and only if, at each point of M2, there exists a local coordinate system (u, v) and a unit speed curveγon Qε2, such thatφ(u,v)=(Cε(u cosθ)γ(v)+Sε(u cosθ)γ(v)×γ'(u),cos(u sinθ),sin(u sinθ)), whereIn the last chapter, we consider hypersurfaces with constant sectional curvature in higher dimentional product manifolds Sn×S and Hn×S. In the case of n≥4, we get two classification theroems: Theorem 4.4.1 Letφ:MKn→Sn×S(n≥4),be an isometric immersion of constant sectional curvature K,then K≥1.Moreover,(ⅰ)If K=1 thenφ(M1n)is an open subset of a slice Sn×{s0}(s0∈S);(ⅱ)If K>1 thenφ(MKn) is an open subset of a hypersurface of rotation given by Theorem 4.2.2.Theorem 4.4.2 Letφ:MKn→Hn×s(n≥4),be an isometric immersion of constant sectional curvature K,then K≥-1.Moreover,(ⅰ)If K=-1 thenφ(M-1n)is an open subset of a slice Hn×{s0},(s0∈S);(ⅱ)If K∈(-1,0)thenφ(MKn) is an open subset of a hypersurface of rotation given by Theorem 4.2.1-(ⅱ);(ⅲ)If K=0 then (a)φ(MKn) is an open subset of M0n-1×S,where M0n-1 is a hypersurface of Hn; (b)φ(MKn) is an open subset of a hypersurface of rotation given by Theorem 4.2.1-(ⅲ)-a;(ⅳ)If K>0 thenφ(MKn) is an open subset of the hypersurface of spherical rotation given by Theorem 4.2.1-(ⅳ).