The Funk-Finsler Structure in the Constant Curvature Spaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866929726496440320 |
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| author | Kumar, Ashok Shah, Hemangi Madhusudan Tiwari, Bankteshwar |
| author_facet | Kumar, Ashok Shah, Hemangi Madhusudan Tiwari, Bankteshwar |
| contents | 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 $-\frac{1}{4}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_16149 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Funk-Finsler Structure in the Constant Curvature Spaces Kumar, Ashok Shah, Hemangi Madhusudan Tiwari, Bankteshwar Differential Geometry 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 $-\frac{1}{4}$. |
| title | The Funk-Finsler Structure in the Constant Curvature Spaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2502.16149 |