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Revisiting the foundations of Barbilian’s metrization procedure. (English) Zbl 1221.53040

The authors prove a natural extension of one of Barbilian’s theorems [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.

MSC:

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
51K05 General theory of distance geometry

Citations:

Zbl 0112.37201
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References:

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