Geodesics and curvature of Möbius invariant metrics
A region \(\Omega\) on the Riemann sphere \(\hat{\mathbb{C}}\) is called a quasi-hyperbolic region if \(\hat{\mathbb{C}}\setminus \Omega\) contains at least two points. In this paper the authors examine the Ferrand metric and the Kulkarni-Pinkall metric on \(\Omega\). The Ferrand metric \(\varphi_{\Omega}(z)|dz|\) is defined by \[ \varphi_{\Omega}(z)=\sup_{a,b\in \Omega^{c}}\tau_{ab}(z) \;\text{for} \;z\in \Omega \cap \mathbb{C}, \] where \(\tau_{ab}(z)=|a-b|/(|z-a||z-b|)\) is the density of the quasi-hyperbolic metric on the region \(\hat{\mathbb{C}}\setminus \{a,b\}\). The Kulkarni-Pinkall metric \(\mu_{\Omega}(z)|dz|\) is defined by \[ \mu_{\Omega}(z)=\inf\{\lambda_{D}(z); z\in D\subset\Omega, \;D \;\text{is a disk on} \;\hat{\mathbb{C}} \} \;\text{for} \;z\in \Omega \cap \mathbb{C}, \] where \(\lambda_{D}(z)\) is the density of the hyperbolic metric on \(D\). It is shown that these metrics are calculated in terms of Euclidean diameters and circumdiameters. One of the main results in this paper is that circular geodesics are also geodesics for these two metrics. Another result is that every Kulkarni-Pinkall isometry between two regions with a certain property, especially nonsimply and nondoubly connected regions, is a Möbius transformation. The other results concern the generalized Gaussian curvatures of these two metrics. The authors study the possible values of the curvatures of these metrics, and determine the quasi-hyperbolic regions on which the curvatures are constant. For many fundamental regions such as strips, infinite sectors, the punctured disk and so on, these metrics and the generalized Gaussian curvatures are presented exactly.
scientific article; zbMATH DE number 5541249