Local isometries of compact metric spaces

Abstract:

By local isometries we mean mappings which locally preserve distances. A few of the main results are: 1. For each local isometry $f$ of a compact metric space $(M,\rho )$ into itself there exists a unique decomposition of $M$ into disjoint open sets, $M = M_0^f \cup \cdots \cup M_n^f$, $(0 \leqslant n < \infty )$ such that (i) $f(M_0^f) = M_0^f$, and (ii) $f(M_i^f) = M_{i - 1}^f$ and $M_i^f \ne \emptyset$ for each $i, 1 \leqslant i \leqslant n$. 2. Each local isometry of a metric continuum into itself is a homeomorphism onto itself. 3. Each nonexpansive local isometry of a metric continuum into itself is an isometry onto itself. 4. Each local isometry of a convex metric continuum into itself is an isometry onto itself.