The Flat Plane and a Constructive Proof of Minding's Theorem

Abstract

Minding's most celebrated result is his namesake theorem of 1839 which established that all surfaces having the same constant curvature must be locally isometric. Today, Minding's theorem is a staple in differential geometry textbooks. But, to the best of our knowledge, all published proofs of it, inclusive of Minding's original argument are existential in nature. In this note, we give a constructive proof of Minding's theorem in the flat case. The proof requires only some basic facts about harmonic functions and complex analytic functions.