The Properties Of Two Class Of Exponential Finsler Metrics

The purpose of this paper is to study a special kinds of (a,β)-metrics—the expo-nential Finsler metrics in the form F=aexp((?)), where α=(?) is a Riemannian metric and β=bi(x)yi is a1-form on the manifold. We discussed the conditions and properties of locally dually flat exponential Finsler metric and locally projectively flat exponential Finsler metric. Firstly, we obtained the necessary and sufficient conditions that the exponential Finsler metrics are locally dually flat. Based on these conditions, we proved that an exponential Finsler metric is a locally dually flat Finsler metric if and only if it is a Minkowski metric when a is locally projectively flat. Further, we characterize locally dually flat exponential Finsler metric with isotropic S-curvature. We obtainde the conditions that the dually flat exponential Finsler metric is of isotropic S-curvature. Finally, we studied locally projectively flat exponential Finsler metrics. We discussed the conditions that an exponential Finsler metric to be Berwald metric. We also proved that an exponential Finsler metric is projectively flat if and only if β is closed.