The Funk-Finsler Structure in the Constant Curvature Spaces

In this paper, we { 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 3/2, in spherical space bounded below by 3/2, and in Euclidean case it is identically equal to 3/2. Further, we show that the flag curvature of the Funk-Finsler metric in hyperbolic space is bounded above by -1/4, in spherical space bounded below by -1/4, and in Euclidean case it is identically equal to -1/4.