On a class of locally projectively flat Finsler metrics

Locally projectively flat Finsler metrics compose an important group of metrics in Finsler geometry. The characterization of these metrics is the regular case of the Hilbert's Fourth Problem. In this paper, we study a class of Finsler metrics composed by a Riemann metric alpha = root a(ij)(x)y(i)y(j) and a 1-form beta = b(i)(x)y(i) called general (alpha, beta)-metrics. We classify those locally projectively flat when alpha is projectively flat. By solving the corresponding nonlinear PDEs, the metrics in this class are totally determined. Then a new group of locally projectively flat Finsler metrics is found.