关于广义m-quasi-Einstein流形的研究

发布时间：2018-04-12 12:41

本文选题：广义m-quasi-Einstein流形 + m-quasi-Einstein流形　；参考：《郑州大学》2017年博士论文

【摘要】：Einstein流形是黎曼几何中的重要研究对象,作为Einstein流形的推广,加权拟Einstein型流形越来越受到人们的广泛关注.本文围绕广义m-quasi-Einstein流形,研究其在典型几何条件下的分类及性质.主要结果如下:第一,完全分类了具有平行Ricci张量的非平凡广义m-quasi-Einstein流形.然后,在更广泛的曲率条件下,研究了具有常Ricci曲率的广义m-quasi-Einstein流形,并在适当假设下,得到此类流形一定是Einstein的.作为推论,我们得到,m=1时,齐性的真的广义m-quasi-Einstein流形也一定是Einstein的.第二,研究了具有常数量曲率的广义m-quasi-Einstein流形的刚性现象.另外,我们借助等参函数理论,研究了常数量曲率的m-quasi-Einstein流形的刚性,得到此类流形的数量曲率只能取得有限个特定的值并能明确表出,其中每个值都有典型的例子可以达到.第三,研究了具有典型几何结构的广义m-quasi-Einstein流形.首先,讨论了三维齐性流形上广义m-quasi-Einstein结构的存在唯一性,得到三维齐性流形中只有空间形式蕴含真的广义m-quasi-Einstein结构;其次,得到了具有warped乘积结构的广义m-quasi-Einstein流形的若干性质.
[Abstract]:Einstein manifold is an important research object in Riemannian geometry. As a generalization of Einstein manifold, weighted quasi Einstein manifold has attracted more and more attention.In this paper, the classification and properties of generalized m-quasi-Einstein manifolds under typical geometric conditions are studied.The main results are as follows: first, the nontrivial generalized m-quasi-Einstein manifolds with parallel Ricci Zhang Liang are completely classified.Then, the generalized m-quasi-Einstein manifold with constant Ricci curvature is studied under the condition of wider curvature, and it is obtained that the manifold must be Einstein under proper assumptions.As a corollary, we obtain that a homogeneous true generalized m-quasi-Einstein manifold must also be Einstein's.Secondly, the rigidity of generalized m-quasi-Einstein manifolds with constant scalar curvature is studied.In addition, with the help of isoparametric function theory, we study the rigidity of m-quasi-Einstein manifolds with constant scalar curvature. It is obtained that the scalar curvature of such manifolds can only obtain a finite number of specific values and can be clearly expressed, each of which can be achieved by a typical example.Thirdly, the generalized m-quasi-Einstein manifolds with typical geometric structures are studied.Firstly, the existence and uniqueness of generalized m-quasi-Einstein structure on three-dimensional homogeneous manifolds are discussed, and the existence and uniqueness of generalized m-quasi-Einstein structures in three-dimensional homogeneous manifolds are obtained. Secondly, some properties of generalized m-quasi-Einstein manifolds with warped product structures are obtained.
【学位授予单位】：郑州大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O186.12

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