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.