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This survey article mainly introduces some important research progresses on the smooth solutions of Hilbert’s fourth problem in the regular case, which we call projectively flat Finsler metrics. We characterize and classify projectively flat Finsler metrics of constant flag curvature. We also discuss and classify projectively flat Finsler metrics with isotropic S-curvature. In particular, we study and characterize projectively flat Randers metrics, square metrics, (α, β)-metrics, and their curvature properties. |
