This is the paper of Su, appeared on American Mathematical Monthly, Number 9, Vol. 104, 1997.
I don’t know about this concept. In fact the metric map is the nonexpansive map.
A continuous function between two metric spaces that does not increase distance, i.e.
There are many names for this concept: metric map, nonexpansive map, Lipschitz map with constant , short map, weak contraction, nonexpanding map.