Busemann spaces of Aleksandrov curvature bounded above
Valerii N. Berestovskii
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Submission date: 03. Oct. 2001
published in: Algebra i analiz, 14 (2002) 5, p. 3-18
with the following different title: Busemann spaces with upper-bounded Aleksandrov curvature
MSC-Numbers: 53C20, 53C23, 53C70
Keywords and phrases: aleksandrov space of curvature bounded above, busemann g-space
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In this paper we prove the following main result:every Busemann G-space with Aleksandrov curvature locally bounded from above is Riemannian -manifold (with -atlas in which the components of metric tensor are continuous). Previously we find a necessary and sufficient conditions for isometricity of a metric space to (finite- or infinitedimensional) Euclidean space or unit sphere in Euclidean space. Also we prove that for locally compact geodesically complete inner metric space Mof Aleksandrov curvature locally bounded from above, the tangent space defined as O-cone over space of directions to M at any point is isometric to Gromov tangent cone defined as Gromov-Hausdorff limit of scaled space M with the base point x.