Some Properties Of The Space In The Form Of Map-like Hypersurfaces

Submanifold is an important research field of differentiable manifold, and hypersurface is the most significant and simplest submanifold. In this thesis, the author studies the geometric properties of the graph-like hypersurfaces in sphere, hyperbolic space and Funk metric spaces. The necessary and sufficient condi-tions for the graph-like hypersurfaces in sphere and hyperbolic space are totally geodesic or minimal are given. And a necessary and sufficient condition for the graph-like hypersurface of the Funk metric spaces is locally projectively flat is given. Main conclusions are as follows:Theorem3.4Let i:Mnâ†'Sn+1(1),(x1,…,xn)â†'(x1,…, xn, f(x1.…, xn)) be a graph-like hypersurface. Then Mn is totally geodesic if and only if f satisfies the following equation: where (x1,…, xn) is the local coordinate of any point x in Mn,f is a C2function about x1,…,xn.Theorem3.5Let i:Mnâ†'Sn+1(1),(x1,…, xn)â†'(x1,…,xn, f(x1,…,xn)) be a graph-like hypersurface. Then Mn is minimal if and only if f satisfies the following equation: where (x1,…, xn) is the local coordinate of any point x in Mn,f is a C2function about x1,…,xn.Theorem4.5Let i:Mnâ†'Hn+1,(x1,…,xn)â†'(x1,…,xn,f(x1,…,xn)) be a graph-like hypersurface. Then Mn is totally geodesic if and only if f satisfying: where (x1,…, xn) is the local coordinate of any point x in Mn, f is a C2function about x1,…,xn, c0,c1,…,cn are constant.Theorem4.9Let i:Mnâ†'Hn+1,(x1,…,xn)â†'(x1,…,xn,f(x1,…,xn)) be a graph-like hypersurface. If f satisfying: then Mn is minimal, where (x1,…, xn) is the local coordinate of any point x in Mn, f is a C2function about x1,…, xn, c0, c1,…, cn are constant.Theorem5.4Let θ be a Funk metric on a n+1-dimensional smooth manifold N, let i:Mnâ†'N,(x1,…,xn)â†'(x1,…,xn, f(x1,…,xn)) be a graph-like hypersurface. Then F=i*θ is locally projectively flat if and only if f satisfying: where (x1,…,xn) is the local coordinate any point x in Mn, f is a function about...