Ex. 2, Chap. 6 in do Carmo's Riemannian Geometry says $\mathrm{T}^2=\mathbf{x}(\mathbb{R}^2)\subset \mathrm{S}^3\subset \mathbb{R}^4$ is a torus with sectional curvature zero in the induced metric, where $\mathbf{x}:\mathbb{R}^2\to\mathbb{R}^4$ is given by $$\mathbf{x}(\theta, \varphi) = \frac{1}{\sqrt{2}}(\cos\theta, \sin\theta,\cos\varphi,\sin\varphi)$$

Since $\mathrm{T}^2$ is two-dimensional, the sectional curvature is indeed the Gaussian curvature, which is intrinsic. But in the usual case $\mathrm{T}^2\subset \mathbb{R}^3$ we know there are points with positive and negative Gaussian curvature in $\mathrm{T}^2$, and this is different from the claim of the exercise.

I am confused why the curvatures in these two cases are different. Since the metrics induced from $\mathrm{S}^3$ and $\mathbb{R}^3$ are the same, the Gaussian curvature should coincide in these two immersions.


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