芬斯勒流形的刚性及李群上的测地向量

发布时间：2018-04-25 13:18

本文选题：芬斯勒流形 + 旗曲率　；参考：《南京师范大学》2017年硕士论文

【摘要】：本文包含两部分研究内容.第一部分我们主要研究了具有严格负旗曲率和几乎常S-曲率的芬斯勒流形以及具有严格负纯量曲率的芬斯勒流形的几何性质.我们主要得到了当芬斯勒流形上的一些非黎曼量(例如:平均嘉当挠率,Matsumoto-挠率等)满足一定的增长条件时的一些刚性结果.主要结论如下:定理3.1.1假设(M,F)是具有几乎常S-曲率的n维完备芬斯勒流形,且其旗曲率K ≤ -α ( α为任意固定正常数).若F的平均嘉当挠率Ⅰ是以次临界指数(?)增长,则F是黎曼的.定理3.2.1假设(M,F)是一个n(n≥3)维完备芬斯勒流形,且其纯量旗曲率K ≤ -α( α为任意固定正常数).若F的Matsumoto-挠率My是以次临界指数(?)增长,则F 一定是Randers度量.特别地,如果F是一个纯量旗曲率满足K ≤ -(?)的紧致芬斯勒流形,则F 一定是Randers度量.测地向量对于研究具有芬斯勒度量的李群上的测地线具有重要意义.因此,在论文的第二部分,我们主要研究了具有左不变Kropina度量与Matsumoto度量的3维连通李群上的测地向量.我们选择一组适当的基底,测地向量就可以用它在这组基下的分量很好地刻画.我们主要得到如下结果:定理4.1.1假设G是3维幺模连通李群,F是G上由黎曼度量a和向量场X = εe1(0 ε 1)定义的左不变Kropina度量,即F(x,y)=ax(y,y)/ax(X,y).则y =y1e1+y2e2+y3e3 ∈g是测地向量当且仅当当y1,y2,y3满足以下方程组:(λ2 -λ3)y2Y3 = 0,2(λ3 - λ1)y12y3+λ1y3|y|2= 0,2(λ1 - λ2)y12y2 - λ1y2|y|2 = 0.其中|y|2 =y12+y22+y32, {λi}是李群G的李代数在一组适当选取的基底{ei}下的结构常数.特别地,如果λ1=λ2 =λ3≠0,则y∈g是测地向量当且仅当y∈Span{e1}.
[Abstract]:This paper consists of two parts. In the first part, we mainly study the ffin manifolds with strictly negative flag curvature and almost constant S- curvature, and the geometric properties of the ffin manifolds with strictly negative pure curvature. We have mainly obtained some non Riemann quantities on the fend manifold (for example, the mean value of the warp torsion, Matsumoto- torsion). The main conclusions are as follows: theorem 3.1.1 hypothesis (M, F) is an n-dimensional complete fnsle manifold with almost constant S- curvature, and its flag curvature is K < - alpha (alpha is arbitrary fixed number). If the average degree of torsion of F is the sub critical exponent (?) growth, then F is Riemann's theorem 3.2.1. It is assumed that (M, F) is a n (n > 3) Vee Bambafensler manifold and its pure scalar curvature K < - alpha (alpha is any fixed normal number). If the Matsumoto- torsion My of F is increased by the sub critical index (?), then F must be a Randers measure. S measurement. The geodesic vector is of great significance to the geodesics on Li Qun, which has a measure of the fin. Therefore, in the second part of the paper, we mainly study the geodesic vector on 3 dimensional connected Li Qun with the left invariant metric and the Matsumoto measure. The components under this group are well depicted. We mainly get the following results: theorem 4.1.1 assumes that G is 3 dimensional Unid connected Li Qun, and F is the left invariant Kropina metric defined by the Riemann metric a and the vector field X = E1 (1), i.e. F (x, y) =ax. Equations: ([lambda 2 - lambda 3) y2Y3 = 0,2 ([lambda 3 - lambda 1) y12y3+ lambda 1y3|y|2= 0,2 (lambda 1 - 2) y12y2 - 1y2|y|2 = 0. in which |y|2 =y12+y22+y32, {[lambda] i} is a structural constant under a set of appropriately selected base {ei}, especially if lambda [lambda] 2 = 3 3 0

【学位授予单位】：南京师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O186.1

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