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

From the point of view of Hilbert's fourth problem, Finsler metrics on an open subset of Rn with positive geodesics that are straight lines are known as locally projectively flat Finsler metrics. In this article, we have studied such projectively flat (cx, /3)-metrics in the form of the special exponential Finsler metric, where cx is a Riemannian metric and /3 is a differential 1-form. We found that the special exponential metric is locally projectively flat if and only if cx is locally projectively flat and /3 is parallel with respect to cx. Furthermore, we obtained the flag curvature and proved that the special exponential metric is locally Minkowskian.