Hypersurfaces With Quasi-Parallel Second Basic Form In Spherical Space

Let x:Mn→ Sn+1 be isometric immersion of Riemannian manifold into an unit sphere space,g and B be Mobius metric and Mobius second fundamental form of x respectively,R is a curvature tensor induced by g.In this thesis,a hypersurface satisfying the condition Q is studied,and the following main theorems are obtained:Theorem 1 Let x:Mn→Sn+1 be an immersed umbilicfree hypersurface with quasi-parallel Mobius second fundamental form.if the Mobius form Φ parallel,then Mn is Mobius equivalent to an open part of the following hypersurfaces:(ⅰ)the torus Sk(a)× Sn-k(?),there 1 ≤k≤n-1;(ⅱ)the image of σ of the standard cylinder Sk(a)× Rn-k in Rn+1,where 1 ≤k≤n-1(ⅲ)the image of τ of the standard cylinder Sk(a)× Hn-k(?)in Hn+1,where 1 ≤k ≤n-2;(iv)CSS(p,q,a).Theorem 2 Let x:Mn→ Sn+1 be an immersed umbilicfree hypersurface with quasiparallel Mobius second fundamental form.if x has only two different Mobius principal curvatures,then Mn is Mobius equivalent to an open part of the following hypersurfaces:(1)the standard torus Sk(r)× Sn-k(?);(2)image of a rotating hypersurface in Rn+1 under the σ;(3)image of a rotating hypersurface in Hn+1 under the τ;(4)x(M)=σ(Γ×Rn-1),Γ(?)R2,where Γ is any smooth curve with non-constant curvature in R2;(5)for any negative constant a,X(M)=π(Hn-1(?)×Γ),Γ(?)S2(l/(?)),where Γ is any smooth curve with non-constant geodesic curvature in S2 1/(?));(6)for any negative constant a,X(M)=π(Γ× Sn-1(1/(?))),Γ(?)H2(1/(?)),whereΓ is any smooth curve with non-constant geodesic curvature inH2(1/(?)).Theorem 3 If Mn is quasi parallel hypersurface isotropic to the Blaschke tensor,then Mn is local Mobius equivalent to the Clifford torus Sk((?))× Sn-k((?)).