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arXiv · 1505.06936

On projectively flat Finsler spaces

T. Q. Binh·D. Cs. Kertész·L. Tamássy
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Abstract

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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T. Q. Binh, D. Cs. Kertész, L. Tamássy. 2015-05-26. On projectively flat Finsler spaces. https://arxiv.org/abs/1505.06936

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