The Riemann mapping theorem (cf e.g. <http://en.wikipedia.org/wiki/Riemann_mapping_theorem>) essentially guarantees the existence of a biholomorphic mapping of a simply connected, open subset of the complex plane onto the unit disk.


**Questions:** 


* Are there any results known about length-preserving mappings from simply connected, open subsets of the complex plane to non-congruent target-subsets of the complex plane?
* What are the necessary and sufficient conditions, under which such mappings exist?
* Are any analogues of the Schwarz-Christoffel mapping (cf e.g. <http://en.wikipedia.org/wiki/Schwarz%E2%80%93Christoffel_mapping>) known for such length-preserving mappings?


Edit: 


**Question under relaxed conditions:**


* are there non-isometric mappings between two simply connected regions with equal perimeter of finite length, which preserve the length of a finite collection of coordinate lines?  

A different way of stating the relaxed problem would be that the source-region is partitioned into a finite collection of simply connected sub-regions, whose interiors do not intersect and the question is, whether non-isometric mappings exist, for which the length of the boundary of every sub-region's image equals the length of the boundary of the corresponding original region.



Length-preserving Analogue of Riemann's Mapping Theorem
Any length preserving map from one plane domain to another must be also conformal.
Because the angle of a small triangle can be found if you know the sides.
This observation solves all your questions, because if a conformal map is also
length preserving, than the derivative must have constant absolute value one, thus it is
of the form $z\mapsto \lambda z+c$ where $|\lambda|=1$, so it is a "roto-translation".


