Hilbert's fourth problem in the constant curvature setting

Abstract

Hilbert's fourth problem seeks the classification of metric geometries where straight lines are shortest paths. Its regular case identifies the projectively flat Finsler manifolds. This broader framework breaks the equivalence between projective flatness and constant curvature that holds in the Riemannian setting, creating a more intricate classification problem. This paper resolves the long-standing question of how the local structure determines the global topology for such manifolds of constant flag curvature, where flag curvature is the natural generalization of Riemannian sectional curvature. We derive explicit distance formulas for all cases of constant flag curvature. For non-positive constant curvature, we establish a global classification of forward complete manifolds, a uniqueness theorem for forward complete metrics, and a characterization of maximal domains of metrics where exotic examples are constructed. For positive constant curvature, we prove a maximum diameter theorem and show the completion of such manifold is a sphere. A fundamental connection is revealed between Sobolev space nonlinearity and backward incompleteness. This work provides a complete characterization of the global geometry for the regular case of Hilbert's fourth problem with constant flag curvature.