Boundedly isometric but not isometric spaces (Q1334411)

Two metric spaces \((M_ i, d_ i)\), \(i=1,2\), are called boundedly isometric if for every bounded set \(A_ 1\subset (M_ 1, d_ 1)\) there is a bounded set \(A_ 2\subset (M_ 2, d_ 2)\) isometric to \(A_ 1\) and if for every bounded set \(A'_ 2\subset (M_ 2, d_ 2)\) there is a bounded set \(A'_ 1\subset (M_ 1, d_ 1)\) isometric to \(A'_ 2\). The author shows that there are boundedly isometric, smooth, complete surfaces in \(\mathbb{R}^ n\) which are not isometric.