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The Funk-Finsler Structure in the Constant Curvature Spaces

Ashok Kumar, Hemangi Madhusudan Shah, Bankteshwar Tiwari

TL;DR

This work identifies and analyzes the infinitesimal Funk-Finsler structure in spaces of constant curvature, presenting a Randers representation $\mathcal{F}_{\epsilon}=\alpha+\beta$ with explicit $\alpha$ and closed $\beta$ for $\epsilon\in\{−1,0,1\}$. It computes the spray coefficients, $S$-curvature, Riemann curvature, Ricci curvature, and flag curvature, proving projective flatness and establishing sharp curvature bounds: $S$-curvature bounds depend on $\epsilon$ (equality only in the Euclidean case) and flag curvature bounds satisfy $K\le -\tfrac{1}{4}$ (hyperbolic), $K\ge -\tfrac{1}{4}$ (spherical), and $K\equiv -\tfrac{1}{4}$ (Euclidean). The paper also provides a Zermelo navigation description of the Funk-Finsler metric on the disc and includes an appendix with the spherical distance formulas essential to the spherical case. Together, these results advance Hilbert-like investigations in constant-curvature Finsler geometries by making the infinitesimal structure explicit and computable.

Abstract

In this paper, we {\it find} the infinitesimal structure of Funk-Finsler metric in spaces of constant curvature. We investigate the geometry of this Funk-Finsler metric by explicitly computing its $S$-curvature, Riemann curvature, Ricci curvature, and flag curvature. Moreover, we show that the $S$-curvature of the Funk-Finsler metric in hyperbolic space is bounded above by $\frac{3}{2}$, in spherical space bounded below by $\frac{3}{2}$, and in Euclidean case it is identically equal to $\frac{3}{2}$. Further, we show that the flag curvature of the Funk-Finsler metric in hyperbolic space is bounded above by $-\frac{1}{4}$, in spherical space bounded below by $-\frac{1}{4}$, and in Euclidean case it is identically equal to $-\frac{1}{4}$.

The Funk-Finsler Structure in the Constant Curvature Spaces

TL;DR

This work identifies and analyzes the infinitesimal Funk-Finsler structure in spaces of constant curvature, presenting a Randers representation with explicit and closed for . It computes the spray coefficients, -curvature, Riemann curvature, Ricci curvature, and flag curvature, proving projective flatness and establishing sharp curvature bounds: -curvature bounds depend on (equality only in the Euclidean case) and flag curvature bounds satisfy (hyperbolic), (spherical), and (Euclidean). The paper also provides a Zermelo navigation description of the Funk-Finsler metric on the disc and includes an appendix with the spherical distance formulas essential to the spherical case. Together, these results advance Hilbert-like investigations in constant-curvature Finsler geometries by making the infinitesimal structure explicit and computable.

Abstract

In this paper, we {\it find} the infinitesimal structure of Funk-Finsler metric in spaces of constant curvature. We investigate the geometry of this Funk-Finsler metric by explicitly computing its -curvature, Riemann curvature, Ricci curvature, and flag curvature. Moreover, we show that the -curvature of the Funk-Finsler metric in hyperbolic space is bounded above by , in spherical space bounded below by , and in Euclidean case it is identically equal to . Further, we show that the flag curvature of the Funk-Finsler metric in hyperbolic space is bounded above by , in spherical space bounded below by , and in Euclidean case it is identically equal to .

Paper Structure

This paper contains 17 sections, 13 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Funk- Finsler metric in the constant curvature spaces
  4. The Papadopoulos-Yamada-Funk metric on spherical disc
  5. Proof:
  6. Proof of the Theorem \ref{['thm1.1']}
  7. Some curvatures of the Funk-Finsler structure on the constant curvature spaces
  8. Spray coefficients and $S$-curvatures of the Funk-Finsler metric $\mathcal{F}_\epsilon$
  9. Proof:
  10. Proof of Theorem \ref{['thm1.3']}:
  11. Proof:
  12. Proof:
  13. Proof:
  14. Proof of Theorem \ref{['thm1.2']}
  15. Proof:
  16. ...and 2 more sections

Key Result

Theorem 1.1

The Funk-Finsler metric $\mathcal{F}_{\epsilon}$ on the disc $\mathbb{D}_E(r)$, centered at origin and radius $0<r\leq r_{\epsilon}$, equipped with a Riemannian metric of constant curvature $\epsilon \in \{-1,0,1\}$ is a Randers metric given by $\mathcal{F}_{\epsilon}(x,\xi )=\alpha_F (x,\xi )+\beta

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Definition 2.1: Finsler structure SSZ; Shiohama-BT
  • Definition 2.2: Randers Metric SSZ
  • Definition 2.3: The Riemann curvature tensor CXSZ
  • Definition 2.4: Flag curvature CXSZ
  • Definition 2.5: Busemann-Hausdorff Volume form in Finsler manifolds SZ; BT
  • Lemma 2.1: W
  • ...and 14 more