Finsler structure of the Apollonian weak metric on the unit disc

Abstract

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 Poincar\'e 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 Poincar\'e 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 $-\infty< K<-1$, in particular, it becomes a Hadamard manifold.