  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.
