Extensions of distance preserving mappings in euclidean and hyperbolic geometry

Abstract.

Suppose that X is a real inner product space of (finite or infinite) dimension at least 2. A distance preserving mapping$f : S \rightarrow X$, where $S \neq \emptyset$ is a (finite or infinite) subset of a finite-dimensional subspace of X, can be extended to an isometry $\varphi$ of X. This holds true for euclidean as well as for hyperbolic geometry. To both geometries there exist examples of non-extentable distance preserving $f : S \rightarrow X$, where S is not contained in a finite-dimensional subspace of X.