Problem
Problem 6.142 Consider the flat torus $T^{2}=\mathbb{R}^{2} / \Gamma$ defined by the lattice
$$
\Gamma=\mathbb{Z} v_{1} \oplus \mathbb{Z} v_{2}, \quad v_{1}=(-\pi, \pi), v_{2}=(0,2 \pi)
$$
(i) Prove that
$$
f(u, v)=(\cos u \cos v, \cos u \sin v, \sin u \cos v, \sin u \sin v)
$$
for $u, v \in(0,2 \pi)$ (see Remark 1.4), is an isometric embedding of $T^{2}$ into the unit sphere $S^{3}$ of $\mathbb{R}^{4}$.
(ii) Prove that the total embedded curvature of $f\left(T^{2}\right)$ in $S^{3}$ is a constant.
Hint Use the Generalised Gauss Theorema Egregium 6.30.