What is the reason that there is no $C^2$ isometric embedding of the flat torus inside $\mathbb{R}^3$? Is there an explicit proof of this fact anywhere? The flat metric must violate some condition for the $C^2$ isometric embedding. And as we know, the condition for a local $C^2$ isometric embedding is given by the Gauss and Codazzi-Mainardi relations. I do not understand how these relations are getting violated by the flat metric on torus. Kindly cite some reference. Thanks in advance.
Edit 1:
I am following this paper: Han and Lin, On the isometric embedding of torus in $\mathbb{R}^3$, Methods and Applications of Analysis 15, pp. 197-204, 2008.
Here, the sufficient conditions for the existence of a global smooth isometric embedding of the torus of genus 1 $\mathbb{T}$ with a Riemannian metric $a$ $(\mathbb{T}, a)$ are given. But these conditions, as can be clearly seen, are given for the existence of the standard embedded torus in $\mathbb{R}^3$, which is too strict. The original question is about the existence of an isometrically embedded torus in $\mathbb{R}^3$, not necessarily the tadard torus. So there must be a different set of sufficient conditions than the ones given in the above reference. Can one of these sufficient conditions be the requirement that the the subset of $\mathbb{T}$ where the Gaussian curvature $K$ of $a$ is positive is non-empty?