论文标题
在兰德斯空间的极端兼容线性连接上
On the extremal compatible linear connection of a Randers space
论文作者
论文摘要
如果Finsler歧管上的线性连接称为与度量的兼容，则其平行传输保留了切线向量的鳍长度。广义的Berwald歧管是配备兼容线性连接的Finsler歧管。由于Finslerian度量标准的兼容性一般并不意味着线性连接的独立性，因此检查Finsler歧管上兼容线性连接存在的第一步是选择最适合寻找的线性连接。在\ cite {v14}中引入了一个合理的选择，称为极端兼容线性连接，该连接在每个点都具有最小范围的扭转。 Randers指标是特殊的Finsler指标，可以写为Riemannian指标和1形式的总和（它们是Riemannian指标的“翻译”）。在本文中，我们研究了与Randers度量的线性连接的兼容性方程。由于兼容的线性连接是由其扭转唯一决定的，因此我们通过将扭转成分作为变量来改变兼容性方程。我们确定这些方程何时具有解决方案，即何时randers空间成为众所周知的Berwald空间，承认兼容线性连接。描述所有这些，我们可以选择最小化属性的极端连接。结果，我们在\ cite {vin1}中获得表征定理：randers空间是一个非瑞曼广告的berwald空间，并且仅当且仅当相对于度量的riemannian部分的扰动项的规范是一个正常数。
A linear connection on a Finsler manifold is called compatible to the metric if its parallel transports preserve the Finslerian length of tangent vectors. Generalized Berwald manifolds are Finsler manifolds equipped with a compatible linear connection. Since the compatibility to the Finslerian metric does not imply the unicity of the linear connection in general, the first step of checking the existence of compatible linear connections on a Finsler manifold is to choose the best one to look for. A reasonable choice is introduced in \cite{V14} called the extremal compatible linear connection, which has torsion of minimal norm at each point. Randers metrics are special Finsler metrics that can be written as the sum of a Riemannian metric and a 1-form (they are "translates" of Riemannian metrics). In this paper, we investigate the compatibility equations for a linear connection to a Randers metric. Since a compatible linear connection is uniquely determined by its torsion, we transform the compatibility equations by taking the torsion components as variables. We determine when these equations have solutions, i.e. when the Randers space becomes a generalized Berwald space admitting a compatible linear connection. Describing all of them, we can select the extremal connection with the norm minimizing property. As a consequence, we obtain the characterization theorem in \cite{Vin1}: a Randers space is a non-Riemannian generalized Berwald space if and only if the norm of the perturbating term with respect to the Riemannian part of the metric is a positive constant.
