2001/09/23 by Zhongmin Shen, Shen, Zhongmin
Mathematics · Physics and Astronomy · #53B40 #53C60 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #math.DG #math.MG #msc:53B40 #msc:53C60
paper · pdf · doi:10.48550/arxiv.math/0109178
26 pages, revised in September, 2001
arxiv created 2001/09/23 · openalex publication_date 2001/09/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is the Hilbert's Fourth Problem to characterize the (not-necessarily-reversible) distance functions on a bounded convex domain in Rn 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). Thus it is a natural problem to study those of constant curvature. In this paper, we study the Hilbert Fourth Problem in the smooth case. We first give a formula for x-analytic projective Finsler metrics F(x,y) of constant curvature K=c using a power series with coefficients expressed in terms of f(y):=F(0, y), h(y):=(1/2)F(x,y)-1Fxk(0, y)yk and c. Then, for any given pair f(y), h(y) and any constant c, we give an algebraic formula for smooth projective Finsler metrics with constant curvature K=c and F(0, y)=f(y) and Fxk(0, y)yk=2f(y)h(y). By these formulas, we obtain several special projective Finsler metrics of constant curvature which can be used as models in Finsler geometry.