On the classifiation of Landsberg spherically symmetric Finsler metrics

DOI10.1142/S0219887821502327arXiv2110.07252OpenAlexW3206026371MaRDI QIDQ6380311

S. G. Elgendi

Publication date: 14 October 2021

Abstract: 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 $ngeq3$, 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.

Full work available at URL: https://doi.org/10.1142/s0219887821502327

Mathematics Subject Classification ID

Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds (58B20) Global differential geometry of Finsler spaces and generalizations (areal metrics) (53C60) Local differential geometry of Finsler spaces and generalizations (areal metrics) (53B40)

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