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Normed Planes in the Cone Type Busemann G-Spaces of Nonpositive Curvature. P. 102–106

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Section: Physics. Mathematics. Informatics




Andreev Pavel Dmitrievich
Institute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov (Arkhangelsk, Russia)
Kolchar Mikhail Aleksandrovich
Institute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov (Arkhangelsk, Russia)


The geometry of the cone type Busemann G-space, that is nonpositively curved G-space 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 G-spaces earlier for proving the Busemann conjecture about the topological manifold of nonpositively curved G-spaces. 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 non-adjugate 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 G-space has a global structure of the normed space? The positive answer to this question allows considering G-spaces of non-positive curvature as almost Finsler manifolds. In this case, the only difference between G-spaces of non-positive curvature and Finsler manifolds is the possible lack of smoothness of norms in the tangent spaces.


Busemann G-space, cone type space, ray, segment, normed plane
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