Preserve one, preserve all

TL;DR

The paper investigates conditions under which functions preserving a small level set of the distance in a complete Riemannian manifold must be isometries, extending understanding of metric space symmetries.

Contribution

It formulates and proves cases of a conjecture linking level set preservation to isometries in complete Riemannian manifolds.

Findings

Functions preserving small distance level sets are isometries in certain cases

Preservation of a single small level set implies strong geometric constraints

The results extend known properties of metric space isometries

Abstract

Isometries of metric spaces $(X,d)$ preserve all level sets of $d$. We formulate and prove cases of a conjecture asserting if $X$ is a complete Riemannian manifold, then a function $f:X \rightarrow X$ preserving at least one level set $d^{-1}(r)$, with $r>0$ small enough, is an isometry.