Revisiting the foundations of Barbilian's metrization procedure
The authors prove a natural extension of one of Barbilian's theorems [\textit{D. Barbilian}, Acad. Repub. Popul. Romine, Studii Cerc. Mat. 11, 7--44 (1960; Zbl 0112.37201)]. The extension supports two ideas: that all distances with constant curvature can be described by Barbilian's metrization principle, and that all Riemannian metrics corresponding to these distances can be obtained with the same unique procedure derived from the main result of the paper. It is shown that the hyperbolic metric of the disk, the hyperbolic metric on the exterior of the disk and the hyperbolic metric on the half-plane can be obtained in a similar way. As well, the authors obtain metrics corresponding to quadratic forms with signature that includes minus. Two oscillant distances are obtained in some subsets of the Lorentz or the Minkowski plane. They provide metrics of generalized Lagrange type on the corresponding subsets. A result concerning the distance induced by a Riemannian metric as a local Barbilian distance, is also proved.
Riemannian metric, generalized Lagrange structure.
scientific article; zbMATH DE number 5925823
scientific article