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The Funk-Finsler structure on the unit disc in the hyperbolic plane

Ashok Kumar, Hemangi Madhusudan Shah, Bankteshwar Tiwari

TL;DR

This work constructs the Funk-Finsler structure on the unit disc within the hyperbolic plane, showing that in the Klein model the metric is a Randers type and a Douglas (non-Berwald) metric. It establishes two equivalent realizations of the Funk-Finsler structure: as a pullback from a Lorentz-Randers metric on the upper sheet of the hyperboloid and as a Randers metric on the hemisphere, highlighting the rich interplay between hyperbolic models. The authors derive explicit formulas for the Finsler data, distance, and several curvature invariants (S-curvature, Riemann, Ricci, and flag curvatures), and they provide the Zermelo navigation data associated with the Randers representation. These results deepen the understanding of Hilbert/Funk geometries in curved spaces and offer concrete tools for exploring geodesics and curvature in Randers-Finsler settings on hyperbolic models.

Abstract

In this paper, we construct the Funk-Finsler structure in various models of the hyperbolic plane. In particular, in the unit disc of the Klein model, it turns out to be a Randers metric, which is a non-Berwald Douglas metric. Further, using Finsler isometries we obtain the Funk-Finsler structures in other models of the hyperbolic plane. Finally, we also investigate the geometry of this Funk-Finsler metric by explicitly computing the S-curvature, Riemann curvature, flag curvature, and Ricci curvature in the Klein unit disc.

The Funk-Finsler structure on the unit disc in the hyperbolic plane

TL;DR

This work constructs the Funk-Finsler structure on the unit disc within the hyperbolic plane, showing that in the Klein model the metric is a Randers type and a Douglas (non-Berwald) metric. It establishes two equivalent realizations of the Funk-Finsler structure: as a pullback from a Lorentz-Randers metric on the upper sheet of the hyperboloid and as a Randers metric on the hemisphere, highlighting the rich interplay between hyperbolic models. The authors derive explicit formulas for the Finsler data, distance, and several curvature invariants (S-curvature, Riemann, Ricci, and flag curvatures), and they provide the Zermelo navigation data associated with the Randers representation. These results deepen the understanding of Hilbert/Funk geometries in curved spaces and offer concrete tools for exploring geodesics and curvature in Randers-Finsler settings on hyperbolic models.

Abstract

In this paper, we construct the Funk-Finsler structure in various models of the hyperbolic plane. In particular, in the unit disc of the Klein model, it turns out to be a Randers metric, which is a non-Berwald Douglas metric. Further, using Finsler isometries we obtain the Funk-Finsler structures in other models of the hyperbolic plane. Finally, we also investigate the geometry of this Funk-Finsler metric by explicitly computing the S-curvature, Riemann curvature, flag curvature, and Ricci curvature in the Klein unit disc.
Paper Structure (26 sections, 19 theorems, 128 equations)

This paper contains 26 sections, 19 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof:
  4. The Funk metric on the Klein unit disc $\mathbb{D}_K(1)$
  5. Proof:
  6. Proof:
  7. Proof:
  8. Proof:
  9. The Realisation of the Funk metric on the Klein unit disc $\mathbb{D}_K(1)$ via the upper sheet of the hyperboloid of two sheets
  10. Proof:
  11. The Realisation of the Funk metric on the Klein unit disc $\mathbb{D}_K(1)$ via the upper hemisphere with radius $r$
  12. Proof:
  13. The Funk-Finsler metric on the Poincaré unit disc in the Poincaré disc $\mathcal{D}_{P}$
  14. Proof:
  15. Proof:
  16. ...and 11 more sections

Key Result

Lemma 2.1

The Busemann-Hausdorff volume form of the Randers metric $F =\alpha + \beta$ is given by, where $dV_\alpha=\sqrt{\det (a_{ij})} \ dx$.

Theorems & Definitions (33)

  • Definition 2.1: Finsler structure SSZ, $\S 1.2$
  • Definition 2.2: Randers Metric SSZ, $\S 1.2$
  • Definition 2.3
  • Definition 2.4: Hilbert metric HHG, Chapter 3, $\S 4$
  • Definition 2.5: Geodesics and Spray coefficients SSZ, $\S 3.1$
  • Definition 2.6: The Riemann curvature tensor CXSZ, $\S 4.1$
  • Definition 2.7: Flag curvature CXSZ, $\S 4.1$
  • Definition 2.8: Busemann-Hausdorff Volume form in Finsler manifolds SZ, $\S 2.2$
  • Lemma 2.1: W, $\S 3$
  • Definition 2.9: S-curvature, SSZ, $\S 5.1$
  • ...and 23 more