Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibi\-li\-ty condition). By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection (preserving the Finslerian length of tangent vectors) is uniquely determined by its torsion. If the torsion is zero then we have a classical Berwald manifold. Otherwise, the torsion is a strange data we need to express in terms of the intrinsic quantities of the Finsler manifold. In the paper we consider the extremal compatible linear connection of a generalized Berwald manifold by minimizing the pointwise length of its torsion tensor. It is a conditional extremum problem involving functions defined on a local neighbourhood of the tangent manifold. In case of a given point of the manifold, the reference element method provides that the number of the Lagrange multipliers equals to the number of the equations providing the compatibility of the linear connection to the Finslerian metric. Therefore the solution of the conditional extremum problem with a reference element can be expressed in terms of the canonical data. The solution of the conditional extremum problem independently of the reference elements can be constructed algorithmically at each point of the manifold. The pointwise solutions constitute a section of the torsion tensor bundle for testing the compatibility of the corresponding linear connection to the Finslerian metric. In other words, we have an intrinsic algorithm to check the existence of compatible linear connections on a Finsler manifold because it is equivalent to the existence of the extremal compatible linear connection.
Finsleroid-Finsler度量是一类重要的奇异（y-局部）Finsler度量。它们是由G. S. 2006年的阿萨诺夫。作为一个特殊的情况下，阿萨诺夫生产的例子兰茨贝格空间的维度至少有三个是不是贝尔瓦尔德类型。他们被称为独角兽[5]。不属于Berwald型的正则（y -全局）Landsberg度量的存在性至今仍是一个开放的问题。本文证明了Asanov的独角兽是广义Berwald流形的一个分支。更精确地说，我们证明了以下定理：Finsleroid-Finsler空间是广义Berwald空间当且仅当Finsleroid荷是常数。特别地，Finsleroid-Finsler空间是Landsberg空间当且仅当它是具有半对称相容线性联络的广义Berwald流形。
Critical Remarks on Finsler Modifications of Gravity and Cosmology by Zhe Chang and Xin Li
我不同意论文arXiv：0806.2184和arXiv：0901.1023v1（发表在《物理快报》上，分别参见B668（2008）453和B676（2009）173）。他们认为这是\textit{"In Finsler manifold, there exists a unique linear connection - the Chern connection ... It is torsion freeness and metric compatibility ... "}。有一些众所周知的结果（例如，H. Rund和R. Miron和M. Anastasiei）证明了在Finsler几何中存在无穷多个由同一度量结构定义的线性联络，并证明了Chern联络和Berwald联络\textbf{are not metric compatible.}。例如，Chern联络（在切丛的基流形上具有零挠率和“弱”相容性）一般不与全空间上的度量结构相容。这导致了芬斯勒旋量和狄拉克算符的定义中的一些额外的困难和复杂性，以及量子引力和非对易/弦/膜/规范理论的进一步推广中的额外问题。我的结论是，标准的物理理论可以推广自然的引力和物质场方程的嘉当和/或任何其他芬斯勒度量兼容的连接。这使我们能够构建更真实的芬斯勒时空模型，各向异性场相互作用和宇宙学。
