本文選題：廣義m-quasi-Einstein流形 + m-quasi-Einstein流形��； 參考：《鄭州大學(xué)》2017年博士論文

【摘要】：Einstein流形是黎曼幾何中的重要研究對(duì)象,作為Einstein流形的推廣,加權(quán)擬Einstein型流形越來(lái)越受到人們的廣泛關(guān)注.本文圍繞廣義m-quasi-Einstein流形,研究其在典型幾何條件下的分類(lèi)及性質(zhì).主要結(jié)果如下:第一,完全分類(lèi)了具有平行Ricci張量的非平凡廣義m-quasi-Einstein流形.然后,在更廣泛的曲率條件下,研究了具有常Ricci曲率的廣義m-quasi-Einstein流形,并在適當(dāng)假設(shè)下,得到此類(lèi)流形一定是Einstein的.作為推論,我們得到,m=1時(shí),齊性的真的廣義m-quasi-Einstein流形也一定是Einstein的.第二,研究了具有常數(shù)量曲率的廣義m-quasi-Einstein流形的剛性現(xiàn)象.另外,我們借助等參函數(shù)理論,研究了常數(shù)量曲率的m-quasi-Einstein流形的剛性,得到此類(lèi)流形的數(shù)量曲率只能取得有限個(gè)特定的值并能明確表出,其中每個(gè)值都有典型的例子可以達(dá)到.第三,研究了具有典型幾何結(jié)構(gòu)的廣義m-quasi-Einstein流形.首先,討論了三維齊性流形上廣義m-quasi-Einstein結(jié)構(gòu)的存在唯一性,得到三維齊性流形中只有空間形式蘊(yùn)含真的廣義m-quasi-Einstein結(jié)構(gòu);其次,得到了具有warped乘積結(jié)構(gòu)的廣義m-quasi-Einstein流形的若干性質(zhì).
[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.
【學(xué)位授予單位】：鄭州大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O186.12

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本文編號(hào)：1739803