On Locally Projectively Flat Finsler Space of a Special Exponential Metric with Constant Flag Curvature

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From the point of view of Hilbert's fourth problem, Finsler metrics on an open subset of $\mathbb{R}^n$ with positive geodesics that are straight lines are known as locally projectively flat Finsler metrics. In this article, we have studied such projectively flat $(\alpha,\beta)$-metrics in the form of the special exponential Finsler metric, where $\alpha$ is a Riemannian metric and $\beta$ is a differential 1-form. We found that the special exponential metric is locally projectively flat if and only if $\alpha$ is locally projectively flat and $\beta$ is parallel with respect to $\alpha$. Furthermore, we obtained the flag curvature and proved that the special exponential metric is locally Minkowskian.