Embedding $2$-Torus in $\Bbb R^3$ A 2-Torus is a 2D surface. Intuitively, it may seem possible to see it as a curved surface in a normal 3D space. For example, a 2-sphere is just a globe. So can we take a small part of a 2-Torus and locally isometrically embed it in an affine $\Bbb R^3$ space?
 A: Eric Wofsey's comment is very relevant: some metrics on the torus allow smooth isometric embeddings into $\mathbb{R}^3$ (we assume the standard metric on $\mathbb{R}^3$ throughout).
Some metrics on the 2-torus do not permit smooth (or even $C^2$, that is, twice-continuously-differentiable) isometric embeddings into $\mathbb{R}^3$.  The relevant nonexistence proofs usually proceed via a curvature argument.

For example, the flat metric on the torus (the one induced from the standard metric on $\mathbb{R}^2$ via the universal covering $\mathbb{R}^2 \to \mathbb{R}^2/\mathbb{Z}^2$) does not admit such an embedding.  Suppose otherwise; the image of $T^2$ is a compact subset of $\mathbb{R}^3$ and hence has some point at maximal distance from the origin.  We may further assume this point to be somewhat isolated, in the sense that an open disc around this point is not all at maximum distance; if we could not do this then we would have embedded the sphere, not the 2-torus.  (A codimension-1 subset of the disc may be at maximum distance, but the whole disc is not.)  Then considering the Gauss curvature of the embedding in one of these discs around a somewhat-isolated point reveals that it must be positive somewhere, contradicting our assumption that we have isometrically embedded the flat 2-torus.

Amazingly, however, the flat torus may be isometrically embedded in $\mathbb{R}^3$ via a $C^1$ map that is not of class $C^2$.  This is a consequence of the Nash-Kuiper embedding theorem.  You can find a beautiful visualization of this at the Hévéa Project, which is further described in the article "Flat tori in three-dimensional space and convex integration" by Borrelli, Jabrane, Lazarus, and Thibert.
