THEOREMA EGREGIUM by Gauss (1827)

The Gauss' curvature is only determinated by the Metrical Fundamental Parameters and their derivation.

One important conclusion for the cartography is:
If two areas can be mapped length-preserving onto each other, thus they have the same Gauss' curvature.

First of all note the inversion of this conclusion:
Two areas with different Gauss' curvature can   n o t   be mapped length-preserving onto each other.