Finsler structure of the Apollonian weak metric on the unit disc
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908554923868160 |
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| author | Pandey, Alok Kumar Kumar, Ashok Tiwari, Bankteshwar |
| author_facet | Pandey, Alok Kumar Kumar, Ashok Tiwari, Bankteshwar |
| contents | In this paper, we {\it find} the Finsler structure of the Apollonian weak metric on the open unit disc in $\mathbb{R}^2$, which turns out to be a Randers type Finsler structure and we call it as Apollonian weak-Finsler structure. In fact the Apollonian weak-Finsler structure is the deformation of the hyperbolic Poincaré metric in the unit disc by a closed $1$-form. As a cosequence, the trajectories of the geodesic of this Apollonian weak-Finsler structure pointwise agrees with the geodesic of hyperbolic Poincaré metric in the open unit disc. Further, we explicitly calculate its $S$-curvature, Riemann curvature, Ricci curvature and flag curvature. It turns out that the $S$-curvature of the Apollonian weak-Finsler structure in the unit disc is bounded below by $\frac{3}{2}$, while its flag curvature $K$ satisfies $\frac{-1}{4}\leq K<2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_18621 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finsler structure of the Apollonian weak metric on the unit disc Pandey, Alok Kumar Kumar, Ashok Tiwari, Bankteshwar Differential Geometry 53B40, 53B60, 53C50 In this paper, we {\it find} the Finsler structure of the Apollonian weak metric on the open unit disc in $\mathbb{R}^2$, which turns out to be a Randers type Finsler structure and we call it as Apollonian weak-Finsler structure. In fact the Apollonian weak-Finsler structure is the deformation of the hyperbolic Poincaré metric in the unit disc by a closed $1$-form. As a cosequence, the trajectories of the geodesic of this Apollonian weak-Finsler structure pointwise agrees with the geodesic of hyperbolic Poincaré metric in the open unit disc. Further, we explicitly calculate its $S$-curvature, Riemann curvature, Ricci curvature and flag curvature. It turns out that the $S$-curvature of the Apollonian weak-Finsler structure in the unit disc is bounded below by $\frac{3}{2}$, while its flag curvature $K$ satisfies $\frac{-1}{4}\leq K<2$. |
| title | Finsler structure of the Apollonian weak metric on the unit disc |
| topic | Differential Geometry 53B40, 53B60, 53C50 |
| url | https://arxiv.org/abs/2509.18621 |