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An investigation of the plane in absolute geometry. (English. Russian original) Zbl 0907.51005
Dokl. Math. 53, No. 3, 384-386 (1996); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 348, No. 4, 449-451 (1996).
The author defines: An absolute geometry is a geometry that satisfies the axioms of the Euclidean geometry without possibly the axiom of parallels. He considers the two-dimensional case and proves that every absolute plane is a two-dimensional Riemann manifold of a constant nonpositive Gaussian curvature \(K\). If \(K=0\) then the plane is Euclidean, and if \(K<0\) then it is the Lobachevskij plane.
MSC:
51F05 Absolute planes in metric geometry
Keywords:
absolute plane
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