Bilet V. Isometric embeddings, curvature and boundedness of pretangent spaces

Relationships between global properties of pretangent spaces and local properties of original metric space are investigated. Criteria of isometric embeddings of pretangent spaces in finite-dimension Euclidean spaces are found. These criteria are infinitesimal versions of the well-known Menger’s and Schoenberg’s theorems. The conditions of existence of fixed pretangent space, that is embedded in Euclidean spaces are also found. The structure of metric spaces with nonpositive and nonnegative Aleksandrov curvature pretangent spaces is described. The criterion of boundedness of pretangent spaces is established.