本文選題：芬斯勒流形 + 旗曲率��； 參考：《南京師范大學(xué)》2017年碩士論文

【摘要】：本文包含兩部分研究?jī)?nèi)容.第一部分我們主要研究了具有嚴(yán)格負(fù)旗曲率和幾乎常S-曲率的芬斯勒流形以及具有嚴(yán)格負(fù)純量曲率的芬斯勒流形的幾何性質(zhì).我們主要得到了當(dāng)芬斯勒流形上的一些非黎曼量(例如:平均嘉當(dāng)撓率,Matsumoto-撓率等)滿足一定的增長(zhǎng)條件時(shí)的一些剛性結(jié)果.主要結(jié)論如下:定理3.1.1假設(shè)(M,F)是具有幾乎常S-曲率的n維完備芬斯勒流形,且其旗曲率K ≤ -α ( α為任意固定正常數(shù)).若F的平均嘉當(dāng)撓率Ⅰ是以次臨界指數(shù)(?)增長(zhǎng),則F是黎曼的.定理3.2.1假設(shè)(M,F)是一個(gè)n(n≥3)維完備芬斯勒流形,且其純量旗曲率K ≤ -α( α為任意固定正常數(shù)).若F的Matsumoto-撓率My是以次臨界指數(shù)(?)增長(zhǎng),則F 一定是Randers度量.特別地,如果F是一個(gè)純量旗曲率滿足K ≤ -(?)的緊致芬斯勒流形,則F 一定是Randers度量.測(cè)地向量對(duì)于研究具有芬斯勒度量的李群上的測(cè)地線具有重要意義.因此,在論文的第二部分,我們主要研究了具有左不變Kropina度量與Matsumoto度量的3維連通李群上的測(cè)地向量.我們選擇一組適當(dāng)?shù)幕?測(cè)地向量就可以用它在這組基下的分量很好地刻畫(huà).我們主要得到如下結(jié)果:定理4.1.1假設(shè)G是3維幺模連通李群,F是G上由黎曼度量a和向量場(chǎng)X = εe1(0 ε 1)定義的左不變Kropina度量,即F(x,y)=ax(y,y)/ax(X,y).則y =y1e1+y2e2+y3e3 ∈g是測(cè)地向量當(dāng)且僅當(dāng)當(dāng)y1,y2,y3滿足以下方程組:(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的李代數(shù)在一組適當(dāng)選取的基底{ei}下的結(jié)構(gòu)常數(shù).特別地,如果λ1=λ2 =λ3≠0,則y∈g是測(cè)地向量當(dāng)且僅當(dāng)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

【學(xué)位授予單位】：南京師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O186.1

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1 陶飛;芬斯勒流形的剛性及李群上的測(cè)地向量[D];南京師范大學(xué);2017年

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本文編號(hào)：1801508