# Isometric Embedding for Surfaces: Classical Approaches and Integrability

## TL;DR

This paper reviews classical methods for isometrically embedding Riemannian surfaces into Euclidean 3-space, focusing on PDE integrability and connecting coordinate-based and moving-frames approaches with recent joint work.

## Contribution

It clarifies the integrability conditions of the embedding PDE and links classical approaches with recent results on the problem.

## Key findings

- The PDE for isometric embedding is integrable under specific conditions.
- Classical approaches can be unified through the analysis of PDE integrability.
- Recent joint work confirms the integrability criteria for the embedding system.

## Abstract

We review classical approaches to the problem of isometrically embedding a Riemannian surface into Euclidean 3-space, including coordinate-based approaches exposited by Darboux and Eisenhart, as well as the moving-frames based approaches advocated by Cartan. In particular, the first approach involves reducing the problem to solving a single PDE; settling the question of when this PDE is integrable by the method of Darboux is the subject of this short note. This is surprisingly easy, since it is related to the analogous question for the isometric embedding system arising from the moving frames approach, and this was accomplished in recent joint work with Clelland, Tehseen and Vassiliou (arXiv:1801.00241).