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Hilbert's fourth problem in the constant curvature setting

Benling Li, Wei Zhao

TL;DR

This work resolves Hilbert’s fourth problem in the regular case for projectively flat Finsler manifolds with constant flag curvature by deriving explicit distance formulas across $K ext{ in}\{0,-1,1 ightarrow$ $d_F(x_1,x_2)$ in closed forms. It provides a comprehensive global classification for $K ightarrow 0,-1$, including forward completeness, uniqueness of metrics on a domain, and the emergence of maximal domains in 2D with exotic topologies. For $K=1$, it proves a sharp diameter bound and shows that the metric completion is a sphere under a natural extension condition, unifying sphere-like behavior with Bryant-type constructions. Finally, it reveals a deep connection between Sobolev space linearity and backward completeness, tying nonlinear Sobolev spaces to the geometry of non-reversible Finsler metrics, thereby linking Hilbert’s problem to nonlinear analysis on metric spaces.

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.

Hilbert's fourth problem in the constant curvature setting

TL;DR

This work resolves Hilbert’s fourth problem in the regular case for projectively flat Finsler manifolds with constant flag curvature by deriving explicit distance formulas across in closed forms. It provides a comprehensive global classification for , including forward completeness, uniqueness of metrics on a domain, and the emergence of maximal domains in 2D with exotic topologies. For , it proves a sharp diameter bound and shows that the metric completion is a sphere under a natural extension condition, unifying sphere-like behavior with Bryant-type constructions. Finally, it reveals a deep connection between Sobolev space linearity and backward completeness, tying nonlinear Sobolev spaces to the geometry of non-reversible Finsler metrics, thereby linking Hilbert’s problem to nonlinear analysis on metric spaces.

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.

Paper Structure

This paper contains 24 sections, 64 theorems, 471 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Preliminaries
  5. Positively homogeneous functions
  6. Elements from Finsler geometry
  7. Projectively flat Finsler metrics
  8. Projectively flat Finsler manifolds with $\mathbf{K}=0$
  9. Global characterization of manifolds
  10. Global existence of metrics
  11. Projectively flat Finsler manifolds with $\mathbf{K} = -1$
  12. Global characterization of manifolds
  13. Global existence of metrics
  14. Projectively flat Finsler manifolds with $\mathbf{K}=1$
  15. Distance and diameter
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(\Omega,F)$ be a projectively flat Finsler manifold of constant flag curvature $\mathbf{K}$ with projective factor $P$. Then for any pair of distinct points $x_1, x_2\in \Omega$, the distance $d_F(x_1, x_2)$ is given by

Figures (2)

  • Figure 1: Evolution of the domain (shaded) within $\Omega$.
  • Figure 2: Evolution of $\mathop{\mathrm{\mathscr{O}}}\nolimits$ (shaded) within $\Omega$.

Theorems & Definitions (142)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Proposition 2.2: BM
  • Lemma 2.3
  • Lemma 2.4: Sh1
  • ...and 132 more