MiS Preprint Repository

In this paper we prove the following main result:every Busemann G-space with Aleksandrov curvature locally bounded from above is Riemannian $C^o$-manifold (with $C^1$-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 $M_x$ defined as O-cone over space of directions to M at any point $x\in M$ is isometric to Gromov tangent cone $T_{x}M$ defined as Gromov-Hausdorff limit of scaled space M with the base point x.