UDC

514.172.2

Authors

Starostina Vera Valeryevna
Institute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov (Arkhangelsk, Rusia)
e-mail: irrefragable@yandex.ru

Abstract

Mutual arrangement of straight lines is very important in the geometry of geodesic spaces. Parallel lines, i.e. the Hausdorff distance between them is finite, are of particular importance in some cases. In general, the parallelism does not imply any special properties. A special case is the so-called convex spaces, or spaces of nonpositive curvature. In this case, the behavior of parallel lines is tightly regulated by a classical Rinow lemma, which states that any two parallel lines in the nonpositive curvature space in the sense of Busemann limit a normed strip, that is a convex subset isometric to a strip between two parallel lines on the plane, equipped with a strictly convex norm. In this paper we prove a generalization of Lemma in the class of spaces of Busemann nonpositive curvature with respect to the distinguished family of segments. Under the distinguished family of segments in a geodesic space is understood a family of Σ, that any two points of space are connected by a unique segment of Σ, and every segment containing in the segment from Σ, also belongs to Σ. Convexity space property with respect to Σ means that in any triangle formed by segments of Σ, the middle line does not exceed half of the base. The main theorem asserts that in the space of nonpositive curvature in the sense of Busemann with respect to the distinguished family of segments any two distinguished lines limit weak normed strip, that is weakly convex subset isometric to the strip between two parallel affine lines in the normed plane. This fact allows us to develop the methods of geometry of spaces of nonpositive curvature in case of G-spaces with a distinguished segments system. The limiting procedure of nonprincipal ultrafilter passage to the limit is used in the proof of the main theorem. Since the existence of a nonprincipal ultrafilter on the set of natural numbers is a consequence of the axiom of choice, one can not consider the proof as constructive. The problem of the proof of Rinow Lemma in a given class of spaces without the use of ultrafilters is closely connected with the existence of Σ-convex normed strip.

Keywords

Rinow lemma, normed strip, weak convexity, nonpositive curvature, distinguished family of segments

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