In this paper, as an application of the inverse problem of calculus of variations, we investigate two compatibility conditions on the spherically symmetric Finsler metrics. By making use of these conditions, we focus our attention on the Landsberg spherically symmetric Finsler metrics. We classify all spherically symmetric manifolds of Landsberg or Berwald types. For the higher dimensions n ≥ 3, we prove that all Landsberg spherically symmetric manifolds are either Riemannian or their geodesic sprays have a specific formula; all regular Landsberg spherically symmetric metrics are Riemannian; all (regular or non-regular) Berwald spherically symmetric metrics are Riemannian. Moreover, we establish new unicorns, i.e. new explicit examples of non-regular non-Berwaldian Landsberg metrics are obtained. For the two-dimensional case, we characterize all Berwald or Landsberg spherically symmetric surfaces.

中文翻译：

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关于 Landsberg 球对称 Finsler 度量的分类

在本文中，作为变分逆问题的一个应用，我们研究了球对称 Finsler 度量的两个相容条件。通过利用这些条件，我们将注意力集中在 Landsberg 球对称 Finsler 度量上。我们对 Landsberg 或 Berwald 类型的所有球对称流形进行分类。对于更高的维度n ≥ 3，我们证明了所有 Landsberg 球对称流形要么是黎曼流形，要么它们的测地线喷雾具有特定的公式；所有常规的 Landsberg 球对称度量都是黎曼度量；所有（常规或非常规）Berwald 球对称度量都是黎曼度量。此外，我们建立了新的独角兽，即获得了非常规非Berwaldian Landsberg 度量的新显式示例。对于二维情况，我们描述了所有 Berwald 或 Landsberg 球对称表面。