The Funk–Finsler Structure on the Unit Disc in the Hyperbolic Plane

In this paper, we construct a Funk–Finsler structure in various models of the hyperbolic plane. In particular, in the unit disc of the Klein model, it turns out to be a Randers metric, which is a non-Berwald Douglas metric. Further, using Finsler isometries we obtain the Funk–Finsler structures in other models of the hyperbolic plane. Finally, we also investigate the geometry of this Funk–Finsler metric by explicitly computing the S-curvature, Riemann curvature, flag curvature, and the Ricci curvature in the Klein unit disc.