Arctic Environmental Research
A peerreviewed openaccess journal
eISSN 26587173 DOI:10.17238/issn25418416
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Section: Physics. Mathematics. Informatics UDC515.16AuthorsAndreev Pavel DmitrievichInstitute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov (Arkhangelsk, Russia) email: pdandreev@mail.ru Kolchar Mikhail Aleksandrovich Institute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov (Arkhangelsk, Russia) email: mishandri@gmail.com AbstractThe geometry of the cone type Busemann Gspace, that is nonpositively curved Gspace X isometric to its space tangent cone KpX, is studied in the paper. Geodesic spaces of this class have a number of significant geometrical properties. The most significant fact is that the group H of positive homotheties hk with the center p acts on X. Andreev P.D. used the cone type Gspaces earlier for proving the Busemann conjecture about the topological manifold of nonpositively curved Gspaces. The basic result of the paper is the theorem stating that any two rays in X beginning at p are contained in some normed plane. In this case the convex subset isometric to the affine plane equipped with strongly convex norm is considered as the normed plane in the geodesic space X. The proof of the theorem is based on the fact that the convex hull of two nonadjugate rays with common beginning in the vertex p is an angle generated by the integration of the fixed segment images with the ends at these rays under the homotheties influence of hk, k > 0. The proven theorem leads to some new problems. First of all, there is a question whether an arbitrary cone type Gspace has a global structure of the normed space? The positive answer to this question allows considering Gspaces of nonpositive curvature as almost Finsler manifolds. In this case, the only difference between Gspaces of nonpositive curvature and Finsler manifolds is the possible lack of smoothness of norms in the tangent spaces.KeywordsBusemann Gspace, cone type space, ray, segment, normed planeReferences
