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On projectively flat Finsler spaces


First we present a short overview of the long history of projectively flat Finsler spaces. We give a simple and quite elementary proof of the already known condition for the projective flatness, and we give a criterion for the projective flatness of a special Lagrange space (Theorem 1). After this we obtain a second-order PDE system, whose solvability is necessary and sufficient for a Finsler space to be projectively flat (Theorem 2). We also derive a condition in order that an infinitesimal transformation takes geodesics of a Finsler space into geodesics. This yields a Killing type vector field (Theorem 3). In the last section we present a characterization of the Finsler spaces which are projectively flat in a parameter-preserving manner (Theorem 4), and we show that these spaces over \({\mathbb {R}}^{n}\) are exactly the Minkowski spaces (Theorems 5 and 6).

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Correspondence to L. Tamássy.

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The publication is supported by the TÁMOP-4.2.2/B-10/1-2010-0024 project. The project is co-financed by the European Union and the European Social Fund.

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Binh, T.Q., Kertész, D.C. & Tamássy, L. On projectively flat Finsler spaces. Acta Math Hung 141, 383–400 (2013).

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