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 32, in spherical space bounded below by 32, and in Euclidean case it is identically equal to 32. Further, we show that the flag curvature of the Funk-Finsler metric in hyperbolic space is bounded above by -14, in spherical space bounded below by -14, and in Euclidean case it is identically equal to -14.