The Local Isometric Embedding of Two-Metrics of Low Differentiability in Euclidean Three Space

TL;DR

This paper proves the local isometric embedding of C1 class metrics into Euclidean 3-space, using PDE solutions and linear algebra, including an example with non-analytic surfaces of constant Gaussian curvature.

Contribution

It establishes the existence of local isometric embeddings for low differentiability metrics in E3, expanding the class of metrics known to be embeddable.

Findings

Existence of local isometric embedding for C1 metrics in E3.

Use of PDE and linear algebra techniques for embedding proof.

Example of a non-analytic surface with Gaussian curvature one.

Abstract

We prove that the isometric embedding of any metric of differentiability class C1 in E3 exists. We use simplified notation for the given metric, namely geodesic parameters, and level parameters for the embedded surface in E3. Central to our discussion will be solutions of initial value problems for two first order non-linear partial differential equations. We also make use of the classical theory of linear algebraic systems. We will prove local isometric embedding. An example is given for which the Gaussian curvature of the metric is equal to one but the embedded surface is non-analytic.