Abstract: In this work, we study the Asanov Finsler metric F=α(β²/α²+gβ/α+1)^1/2exp{(G/2)arctan[β/(hα)+G/2]}, where α=(α_ijyⁱy^j)^1/2 is a Riemannian metric and β=b_iy^j is a 1-form, g∈(−2,2), h=(1−g²/4)^1/2, G=g/h. We give the necessary and sufficient condition for Asanov metric to be locally projectively flat, i.e., α is projectively flat and β is parallel with respect to α. Moreover, we proved that the Douglas tensor of Asanov Finsler metric vanishes if and only if β is parallel with respect to α.

Key words: Exponential Finsler metric, Projectively flat, (α,β)-metrics, Douglas tensor

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