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The Landsberg equation of a Finsler space. (English) Zbl 1508.53078

There are two important types of Finsler spaces: Landsberg spaces and Berwald spaces. Every Berwald space is known to be a Landsberg space. A long standing problem of Finsler geometry asks if there is a Landsberg space which is not a Berwald space (such a Landsberg space is called unicorn). The authors’ main theorem (Theorem 1.2) states that every Landsberg \(\left( \alpha_{1} ,\alpha_{2}\right) \)-space is a Berwald space. Hence, there are no unicorns among the wide class of Landsberg \(\left( \alpha_{1},\alpha_{2}\right) \)-spaces. The authors’ method uses a special system of partial differential equations called Landsberg equations. As a corollary of their main result, the authors prove that the \(S\)-curvature of any Landsberg \(\left( \alpha_{1},\alpha_{2}\right) \)-metric vanishes identically (Theorem 1.1).

MSC:

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C30 Differential geometry of homogeneous manifolds
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References:

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