对偶平坦和共形平坦的(α,β)-度量(英文)

本文主要研究了对偶平坦和共形平坦的(α,β)-度量.利用对偶平坦和共形平坦与其测地线的关系,得到了局部对偶平坦和共形平坦的Randers度量是Minkowskian度量的结论.进一步,推广到非Randers型的情形,我们证明了局部对偶平坦和共形平坦的非Randers型的(α,β)-度量在附加的条件下一定是Minkowskian度量.     

射影平坦Finsler度量的解析构造

利用Hamel关于射影平坦的基本方程,我们导出了Randers度量的λ形变保持射影平坦的充分条件.特别,对一类具有特殊旗曲率性质的Randers度量我们证明了这类度量一定存在保持射影平坦性的λ形变.     

共形平坦的(α,β)-度量

本文主要研究共形平坦的(α,β)-度量.通过共形相关的Finsler度量间其测地系数间的关系,得到了(α,β)-度量是共形平坦的充分必要条件,并构造了若干共形平坦(α,β)-度量的例子.在此基础上,发现共形平坦且具有迷向S-曲率的(α,β)-度量一定是Minkowski度量或Riemann度量.     

具有迷向S曲率Randers度量的几何意义

Riemann流形上的Zermelo航行为Randers度量提供了一个简洁而且清晰的几何背景．在这个背景下D．Bao，C．Robles和Z．Shen对于具有常旗曲率的Randers度量进行了完全分类．这篇论文中，我得到了判定具有特殊曲率性质的Randers度量的两个充分必要条件．从这两个条件出发，我得到了迷向S曲率的Randers度量的几何意义和一系列推论，并且构造了具有迷向S曲率Randers度量的新例子．最后，在Zermelo航行的背景下研究了Berwald型的Raiders度量．     

Riemann流形上的一类标准共形不变度量

任意紧Riemann面上都存在一个仅依赖于共形类且拥有常曲率的度量.Harbermann和Jost用Yamabe算子对应的Green函数在数量曲率为正的局部共形平坦流形上构造了一个标准共形不变度量.在此之后,这类标准共形不变度量被推广到了数量曲率为正的球型CR流形上.进一步的,应用相应的Yamabe算子对应的Green函数可以构造数量曲率为正的球型四元切触流形和数量曲率为正的八元切触流形上类似的标准共形不变张量.在四元切触正质量猜测和八元切触正质量猜测成立的前提下,上述共形不变张量是共形不变度量.文中利用Paneitz算子对应的Green函数在局部共形平坦流形上构造了一类上述标准共形不变张量,并且在一定条件(详见定理3.1)下,该标准共形不变张量进一步为标准共形不变度量.     

射影Ricci平坦的Kropina度量

本文研究和刻画了射影Ricci平坦的Kropina度量.利用Kropina度量的S-曲率和Ricci曲率的公式,得到了Kropina度量的射影Ricci曲率公式.在此基础上得到了Kropina度量是射影Ricci平坦度量的充分必要条件.进一步,作为自然的应用,本文研究和刻画了由一个黎曼度量和一个具有常数长度的Killing 1-形式定义的射影Ricci平坦的Kropina度量,也刻画了具有迷向S-曲率的射影Ricci平坦的Kropina度量.在这种情形下,Kropina度量是Ricci平坦度量.     

一类射影平坦的Finsler度量

本文主要研究由两个Riemann度量和一个1-形式构成的Finsler度量.首先,本文给出这类度量局部射影平坦的等价条件;其次,给出这类度量局部射影平坦且具有常旗曲率的分类情形;最后,构造这类度量局部射影平坦且具有常旗曲率K=-1的例子.     

关于局部对称共形平坦空间中具有常数量曲率的子流形

该文研究了局部对称共形平坦空间中具有常数量曲率的紧致子流形,证明了这类子流形的某些内蕴刚性定理.     

关于共形Berwald的Kropina度量(英文)

本文利用导航数据研究了共形Berwald的Kropina度量.首先利用导航数据刻画了Berwald Kropina度量.在此基础上,本文得到了Kropina度量是共形Berwald度量的一个充分必要条件.进一步,刻画了具有弱迷向旗曲率的共形Berwald Kropina度量的局部结构.     

具有平行平均曲率向量子流形的曲率估计与稳定性

本文估计空间形式中具有平行平均曲率向量子流形上共形度量的数量曲率上界,并利用其研究了具有常平均曲率超曲面的稳定性．     

On some projectively flat finsler metrics in terms of hypergeometric functions

This paper gives an explicit construction of a family of projectively flat Finsler metrics by using hypergeometric functions and a special class of projectively flat Randers metrics.     

The holonomy group of locally projectively flat Randers two-manifolds of constant curvature

In this paper, we investigate the holonomy structure of the most accessible and demonstrative 2-dimensional Finsler surfaces, the Randers surfaces. Randers metrics can be considered as the solutions of the Zermelo navigation problem. We give the classification of the holonomy groups of locally projectively flat Randers two-manifolds of constant curvature. In particular, we prove that the holonomy group of a simply connected non-Riemannian projectively flat Finsler two-manifold of constant non-zero flag curvature is maximal and isomorphic to the orientation preserving diffeomorphism group of the circle.     

Projectively flat Randers metrics with constant flag curvature

 We classify locally projectively flat Randers metrics with constant Ricci curvature and obtain a new family of Randers metrics of negative constant flag curvature. Received: 19 July 2001 / Revised version: 15 March 2002 / Published online: 16 October 2002     

A global classification result for Randers metrics of scalar curvature on closed manifolds

One important problem in Finsler geometry is that of classifying Finsler metrics of scalar curvature. By investigating the second-order differential equation for a class of Randers metrics with isotropic S$S$-curvature, we give a global classification of these metrics of scalar curvature, generalizing a theorem previously only known in the case of locally projectively flat Randers metrics.     

Finsler空间上的Weyl曲率

The Weyl curvature of a Finsler metric is investigated. This curvature constructed from Riemannain curvature. It is an important projective invariant of Finsler metrics. The author gives the necessary conditions on Weyl curvature for a Finsler metric to be Randers metric and presents examples of Randers metrics with non-scalar curvature. A global rigidity theorem for compact Finsler manifolds satisfying such conditions is proved. It is showed that for such a Finsler manifold,if Ricci scalar is negative,then Finsler metric is of Randers type.     

Bach-flat Lie groups in dimension 4

We establish the existence of solvable Lie groups of dimension 4 and left-invariant Riemannian metrics with zero Bach tensor which are neither conformally Einstein nor half conformally flat.     

Conformal metrics of constant mass

We give a new interpretation of the canonical metrics associated to scalar positive locally conformally flat structures introduced in a previous paper. This allows us to extend the definition to not necessarily locally conformally flat structures in low dimensions. Received May 1, 2000 / Accepted May 9, 2000 / Published online September 14, 2000     

Conformal flatness and self-duality of Thurston-geometries

We show which Thurston-geometries in dimensions 3 and 4 admit invariant conformally flat or half-conformally flat metrics.