Vol. 103, No. 2, 1982

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ISSN: 0030-8730
On local isometries of finitely compact metric spaces

Aleksander Całka

Vol. 103 (1982), No. 2, 337–345
Abstract

By local isometries we mean mappings which locally preserve distances. Local isometries which do not increase distances are called nonexpansive local isometries. A few of the main results are:

  1. Let f be a local isometry (nonexpansive local isometry) of a finitely compact metric space (M,ρ) into itself. If for each (some) z M the sequence {fn(z)} is bounded, then there exists a unique decomposition of M into disjoint open sets, M = M0f M1f , such that (i) f maps M0f injectively into itself, and (ii) f(Mi+1f) Mif for each i = 0,1, . Moreover, f maps M0f homeomorphically (isometrically) onto itself.
  2. Let f be a nonexpansive local isometry (local isometry) of a connected (convex) finitely compact metric space (M,ρ) into itself. If for some z M the sequence {fn(z)} is bounded, then f is an isometry onto.

Mathematical Subject Classification 2000
Primary: 54E40
Secondary: 54E45
Milestones
Received: 5 January 1981
Revised: 20 September 1981
Published: 1 December 1982
Authors
Aleksander Całka