On a Class of Curvature Properties of Projectively flat Finsler (α,β) -Metric

Abstract

In this paper, we study a class of Finsler metric in the form \(F=\alpha+\beta+\frac{2 \beta^{2}}{\alpha}-\frac{\beta^{4}}{3 \alpha^{3}}\) where \(\alpha= \sqrt{a_{i j} y^{i} y^{j}}\) is a Riemannian metric, \(\beta=b_{i} y^{i}\) is a 1−form. We obtain a necessary and sufficient condition for \(F\) to be locally projectively flat. Further, we prove that such projectively flat Finsler metrics with the constant flag curvature must be locally Minkowskian.