【作者】 王志张；

【导师】 黄宣国；

【作者基本信息】 复旦大学 ， 基础数学， 2009， 博士

【摘要】 本文主要对于乘积流形上的预给定曲率的问题做了一些研究。我们主要研究了两个问题。第一个是在乘积单位球面上,我们给出自然的由乘积单位球面到高一维球面的嵌入映射,然后考虑了在乘积单位球面上预给定Gauss-Kronecker曲率后,我们所考虑的嵌入的存在性。我们得到了:当乘积单位球面不是由同维数的球面构成的话,在所谓PHC-域上,预给定光滑的曲率函数存在这类嵌入。第二个是在任意乘积流形上,引入了我们称为体积元保持变换的一种度量变换,我们研究了对于流形上预给定的数量曲率,是否存在体积元保持变换将流形上原有度量变为预给定数量曲率的度量。我们得到了:当流形是一个闭流形乘上较小的一段区间时存在这类变换或是一个闭流形乘上任意有限区间,但是预给定的数量曲率比较小的时候也存在这类变换。更多还原

【Abstract】 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.更多还原