We describe the recent developments concerning the rational analogues of the Beckman-Quarles Theorem, and discuss a related result concerning isometric embeddings in Q of subsets of E. 1 – Let E denote the Euclidean d-space, and let Q denote the Euclidean rational d-space. A mapping f : E → E is called ρ-distance preserving if ‖x − y‖ = ρ implies that ‖f(x)−f(y)‖ = ρ. The Beckman Quarles Theore...

Beckman and Quarles proved that a unit distance preserving mapping from a Euclidean space E into itself is necessarily an isometry. In this paper, we give an example of a (non-strictly convex) normed space H for which every unit distance preserving function from H into itself is an isometry. MSC 2000: 51M05, 52A10, 52A20, 52C05, 52C25

L. Paoluzzi constructed a family of compact orientable three-dimensional hyperbolic manifolds with totally geodesic boundary, which were, by construction, closely related to the three-dimensional torus. This paper gives their complete classification up to isometry, and also their isometry groups. The key tool is the so-called canonical decomposition of hyperbolic manifolds.

This paper extends the theory of turbulence of Hjorth to certain classes of equivalence relations that cannot be induced by Polish actions. It applies this theory to analyze the quasi-isometry relation and finite Gromov-Hausdorff distance relation in the space of isometry classes of pointed proper metric spaces, called the Gromov space.