I posted this question recently on math stackexchange (https://math.stackexchange.com/questions/2291382/c2-isometric-embedding-of-the-flat-torus-into-mathbbr3) but have not received any satisfactory answer yet. I am framing a refined version of the problem in the following.

**Question 1** *Is there any smooth, or at least $C^2$, global isometric embedding of the flat torus into $\mathbb{R}^3$?*

We know that the answer is *no*. The standard reason given for the non-existence is that the given flat metric $a_0$ on the torus $\mathbb{T}$ violates the necessary requirement for any global, sufficiently smooth, isometrically embedded compact 2-manifold inside $\mathbb{R}^3$ that, there must be one elliptic point, i.e., a point where the Gaussian curvature of $a_0$ is positive. On the flat torus, there is no such point. Here, we can make a note that the flat torus, however, satisfies the Gauss-Bonnet theorem; in other words, the flat metric $a_0$ is indeed a valid metric on the torus.

This has led me to the following question:

**Question 2 (On the global isometric embedding of a torus into $\mathbb{R}^3$)** Let a sufficiently smooth Riemannian metric $a$ be given on the torus of genus 1, $\mathbb{T}$. $(\mathbb{T}, a)$ is, thus, a sufficiently smooth 2-dimensional Riemannian manifold. Can we pose the statement *the subset of $\mathbb{T}$ where the Gaussian curvature $K$ of $a$ is positive has to be non-empty* as a sufficient condition on the given metric $a$ for the existence of a global isometric embedding of $(\mathbb{T}, a)$ into $\mathbb{R}^3$?

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 $(\mathbb{T}, a)$ are given. But these conditions, as can be clearly seen, are given for the existence of the standard embedded torus in $\mathbb{R}^3$ (the standard illustration of the embedded torus in the literature), which is too strict. The original question is much weaker; it seeks the existence of *some* (sufficiently smooth) isometrically embedded torus inside $\mathbb{R}^3$, not necessarily the standard torus. So there must be a different set of sufficient conditions than the ones given in the above reference. This is the brief motivation that has led me to **Question 2**. Of course, the local sufficient requirements imposed by the Gauss-Codazzi equations must hold. But, interestingly, we need a *valid* second fundamental form $b$ to check the Gauss-Codazzi equations. Taking $b$ as identically zero is a choice. But is it a valid choice? This would then imply that a local, at least $C^2$, isometric embedding is possible. But there seems to be a problem somewhere in this logic.

Thanks in advance for any help or reference!