What is the intrinsic curvature of a flat torus embedded in ℝ⁴?

The minimal codimension for an isometric embedding of the flat torus $T^2$ is $2$, i.e., the minimal ambient dimension is $4 = 2 + 2$. This is a consequence of the Nash embedding theorem: any smooth $n$-dimensional Riemannian manifold admits an isometric embedding into $\mathbb{R}^{n(n+3)/2}$ (for $n=2$, this gives $\mathbb{R}^5$), but the flat torus does better. The explicit Clifford embedding $$(\theta, \phi) \mapsto \frac{1}{\sqrt{2}}(\cos\theta, \sin\theta, \cos\phi, \sin\phi)$$ gives an isometric embedding into $S^3(1/\sqrt{2}) \subset \mathbb{R}^4$, where $S^3(r)$ is the 3-sphere of radius $r$. The **real** question is why $\mathbb{R}^3$ fails. By Gauss's *Theorema Egregium*, Gaussian curvature is an intrinsic invariant. A torus of revolution in $\mathbb{R}^3$ has non-constant curvature — it is positive on the outside and negative on the inside. To have identically zero curvature everywhere, the embedding must leave $\mathbb{R}^3$, as the sum of the sectional curvatures in any 2-plane of $\mathbb{R}^4$ can cancel out. In fact, by a theorem of Tompkins and Chern-Kuiper, any isometric immersion of a flat $n$-manifold into $\mathbb{R}^{2n-1}$ is impossible for $n \geq 2$. For $n=2$, this means no isometric immersion of a flat torus exists in $\mathbb{R}^3$, confirming $\mathbb{R}^4$ as minimal.