Projectively Flat Finsler Metrics of Constant Curvature

It is the Hilbert’s Fourth Problem to characterize the (not-necessarilyreversible) distance functions on a bounded convex domain in R such that straight lines are shortest paths. Distance functions induced by a Finsler metric are regarded as smooth ones. Finsler metrics with straight geodesics said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). In this paper, we study the Hilbert Fourth Problem in the smooth case. We give a formula for x-analytic projective Finsler metrics with constant curvature using a power series with coefficients expressed in terms of F (0, y) and Fxk (0, y)y . We also give a formula for general projective Finsler metrics with constant curvature using some algebraic equations depending on F (0, y) and F xk(0, y)y . By these formulas, we obtain several interesting projective Finsler metrics of constant curvature which can be used as models in certain problems.