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

## TL;DR

This paper provides a constructive proof of Minding's theorem specifically for flat surfaces, utilizing basic harmonic and complex analytic functions, offering a more explicit approach than traditional existential proofs.

## Contribution

It introduces a constructive proof of Minding's theorem in the flat case, contrasting with previous existential proofs, using elementary harmonic and complex analysis tools.

## Key findings

- Constructive proof of Minding's theorem for flat surfaces
- Simplifies understanding of surface isometries in the flat case
- Relies on basic harmonic and complex analytic functions

## 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.