A.D. Alexandrov's problem for Busemann non-positively curved spaces

TL;DR

This paper completes the study of Alexandrov's problem for Busemann non-positively curved spaces, showing that bijections preserving unit distance are isometries in certain geodesically complete spaces.

Contribution

It provides a characterization of isometries in geodesically complete Busemann spaces based on preservation of unit distances, extending previous results.

Findings

Bijections preserving distance 1 are isometries in these spaces.

Characterization applies to geodesically complete, connected at infinity, proper Busemann spaces.

Completes the solution of Alexandrov's problem for this class of spaces.

Abstract

The paper is the last in the cycle devoted to the solution of Alexandrov's problem for non-positively curved spaces. Here we study non-positively curved spaces in the sense of Busemann. We prove that if $X$ is geodesically complete connected at infinity proper Busemann space, then it has the following characterization of isometries. For any bijection $f:X\to X$, if $f$ and $f^{-1}$ preserve the distance 1, then $f$ is an isometry.