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

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