Two Existence Theorems On Product Manifold

The present report mainly studies prescribed curvature problems on product manifolds. We consider two problems.On the product of unit spheres,we give a kind of natural embeddings from the product unit spheres to the unit sphere in which the product of unit spheres can be viewed as a hypersurface.Now the first problem is:for a given positive function on the product of unit spheres,can we find an embedding of this kind such that its Gauss-Kronecker curvature is the given function.We obtain that on so-called PHC-domains,the existence is hold with the hypothesis that the product of unit spheres is not composed by the same dimensional unit sphere.On arbitrary product manifolds,we introduce a class of metric deformations which are called the volume element preserving deformation.Now the second problem is:for given scalar curvatures on some product manifold,can we find some volume element preserving deformation to satisfying that the scalar curvature of deformed metric coincides with the prescribed curvature.We obtain the existence in two cases.In the first case,the product manifold is a closed manifold timing a sufficiently small interval.In the second case, the product manifold is a closed manifold timing a finite interval,but the prescribed curvature should be sufficiently small.