A Proof of the Isometric Embedding Theorem in Three Dimensional Euclidean Space

TL;DR

This paper presents a proof that any two-dimensional metric of class C^1 can be isometrically embedded into three-dimensional Euclidean space, using PDE theory without relying on curvature.

Contribution

It provides a new proof of the isometric embedding theorem in E^3 that does not depend on the metric's curvature, employing first order PDE methods.

Findings

Successful proof of isometric embedding in E^3 for C^1 metrics

Method does not require curvature assumptions

Employs PDE theory for the proof

Abstract

A proof of the isometric embedding of a given two-metric in E^3 of class C^1. The method uses the theory of first order partial differential equations. The curvature of the metric plays no role in the proof.