【作者】 周林峰；

【导师】 莫小欢；

【作者基本信息】 北京大学 ， 基础数学， 2007， 博士

【摘要】 本文主要考虑Finsler几何中的两个重要的几何量: Riemann几何量-旗曲率和非Riemann几何量-Cartan张量.我们利用常旗曲率方程研究了一类具有常旗曲率的Finsler度量,并给出了其局部的一个分类;接着利用Berwald标架证明了这类度量的Cartan张量是有界的.本文主要分为三章:在第一章中,我们介绍了Finsler几何中与以后各章相关的一些基本概念和定理,以便使本文在内容上完备.在第二章中,我们计算了(α,β)度量的Riemann曲率和Ricci曲率在自然坐标系下的局部表达式,并用此公式来讨论具有常旗曲率的一类(α,β)度量F =α(1 +β/α)~p (|p|≥1).我们给出了当p = 2时,此度量具有常旗曲率的充要条件,并证明了它们是局部射影平坦的.由此完成了常旗曲率的度量F = ((α+β)~2)/α的局部分类,解决了由B.Li和Z.Shen提出的一个问题[21].当p = -1时,我们得到了此度量具有常旗曲率的必要条件,而且发现不存在非平凡的常旗曲率的Matsumoto度量.更一般地,当1形式β为闭时,是否存在非平凡的常旗曲率的度量F =α(1 +β/α)~p (|p|≥1)?除了p = 1,2以外,我们给予了否定的回答.在第三章中,我们计算了(α,β)度量的Cartan张量和平均Cartan张量,并给出了它们之间的关系.接着我们继续讨论了度量F =α(1 +β/α)~p的Cartan张量,证明了当p属于区间[1,2)时,其Cartan张量是有界的.然后我们给出了此定理的应用,得出了关于曲率的两个自然的推论.更多还原

【Abstract】 In this dissertation, we mainly discuss two important quantities in Finsler geome-try. One is Riemannian quantity: flag curvature, the other is non-Riemannian quantity:Cartan torsion. We use the equation of constant flag curvature to study a class of Finslermetrics with constant flag curvature and give a local classification; then making use ofBerwald frame we discuss the bound of such metrics’Cartan torsion.This dissertation is mainly made up of following three chapters. In the first chap-ter, we introduce some basic conceptions and theorems in Finsler geometry which arerelative to the next several chapters so that it can be self-contained.In the second chapter, we compute Riemannian curvature and Ricci curvature of(α,β) metrics in local coordinates. Then we apply these formulae to discuss a specialclass (α,β) metrics F =α(1 +β/α)~p (|p|≥1) which have constant flag curvature.We obtain the sufficient and necessary conditions that F = ((α+β)~2)/αhave constant flagcurvature and prove that such metrics must be locally projectively flat. Thus we com-plete their local classification and answer the question of B.Li and Z.Shen [21]. Whenp = -1, we find a necessary condition that flag curvature of F =α~2/(α+β) is constant andprove that there are no non-trivial Matsumoto metrics. Furthermore, we give a negativeanswer whether there are non-trivial metrics F =α(1 +β/α)~p (|p|≥1) of constantflag curvature except for p = 1, 2 whenβis closed.In the last chapter, we compute Cartan torsion and mean Cartan torsion of (α,β)metrics and give the relationship between them. Then we continue to discuss the Cartantorsion of Finsler metrics F =α(1 +β/α)~p (|p|≥1). We prove that when p belongsto an interval [1, 2), Cartan torsion of such metrics is bounded. Using this theorem, weobtain two natural corollaries about curvature.更多还原