I'll explain my idea because it is very likely that this is not the correct name. Suppose you have a surface, tie a string around it and want to move it along one axis (say X axis). What I'm thinking is that if the surface has "constant perimeter" the string would still fit to the surface's shape and move through it. However, if this wasn't the case the string would either not be able to move through the whole surface or just be too loose. A simple example of this is a cylinder where you can wrap a string around it in a circular shape and move it freely through one axis.
I was wondering if the concept of "constant perimeter surface" exists and if there are any resources or you know about more complex examples of it.