On families of Finsler metrics

2025/06/04 by Ísmail Sağlam, Saglam, Ismail, Ken’ichi Ohshika +3 · 1 citation
Physics and Astronomy · #Advanced Differential Geometry Research

paper · pdf · doi:10.48550/arxiv.2506.03749

Abstract

In this paper, we answer some natural questions on symmetrisation and more general combinations of Finsler metrics, with a view towards applications to Funk and Hilbert geometries and to metrics on Teichmüller spaces. For a general non-symmetric Finsler metric on a smooth manifold, we introduce two different families of metrics, containing as special cases the arithmetic and the max symmetrisations respectively of the distance functions associated with these Finsler metrics. We are interested in various natural questions concerning metrics in such a family, regarding its geodesics, its completeness, conditions under which such a metric is Finsler, the shape of its unit ball in the case where it is Finsler, etc. We address such questions in particular in the setting of Funk and Hilbert geometries, and in that of the Teichmüller spaces of several kinds of surfaces, equipped with Thurstonlike asymmetric metrics.

Citations

• On spaces of Euclidean triangles and triangulated Euclidean surfaces
• The earthquake metric on Teichmüller space
• On the Teichmüller space of acute triangles
• Timelike Hilbert geometry of the spherical simplex
• Absolutely continuous curves in Finsler-like spaces
• Minimal stretch maps between Euclidean triangle?
• Tangent spaces of the Teichmüller space of the torus with Thurston's weak metric
• Bijections of geodesic lamination space preserving left Hausdorff convergence
• Timelike Hilbert and Funk geometries
• On Hilbert's fourth problem
• Funk and Hilbert geometries from the Finslerian Viewpoint
• The Funk and Hilbert geometries for spaces of constant curvature
• Harmonic symmetrization of convex sets and of Finsler structures, with applications to Hilbert geometry
• Minimal stretch maps between hyperbolic surfaces
• On an Asymmetrical Metric in the Four-Space of General Relativity

Cited by

• Hilbert metric and Hölder continuity
• On the average scale-invariant Cassinian metric

Related

• On the Finsler stucture of the Teichmüuller metric and Thurston's asymmetric metric
• Harmonic symmetrization of convex sets and of Finsler structures, with applications to Hilbert geometry
• On Projectively Flat Spherically Symmetric Finsler Metrics
• Tangent spaces of the Teichmüller space of the torus with Thurston's weak metric
• Asymmetry in Hilbert's fourth problem