Your definition of the Clifford torus is off. The usual definition of the Clifford torus is the set $(z_1,z_2)\in\mathbb C^2$ in the unit sphere $|z_1|^2+|z_2|^2=1$ with $|z_1|^2=|z_2|^2=\frac 1 2$. This is a square torus isometric to $\mathbb R^2/\Gamma_c$ with $\Gamma_c$ generated by $(2\pi/\sqrt 2, 0), (0, 2\pi/\sqrt 2)$ (not $(2\pi, 0), (0, 2\pi)$) which is isometric to $\mathbb R^2/\Gamma_1$ by a $\pi/4$ rotation.
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Your definition of the Clifford torus is off. The usual definition of the Clifford torus is the set $(z_1,z_2)\in\mathbb C^2$ in the unit sphere $|z_1|^2+|z_2|^2=1$ with $|z_1|^2=|z_2|^2=\frac 1 2$. This is a square torus isometric to $\mathbb R^2/\Gamma_c$ with $\Gamma_c$ generated by $(2\pi/\sqrt 2, 0), (0, 2\pi/\sqrt 2)$ (not $(2\pi, 0), (0, 2\pi)$) which isometric to $\mathbb R^2/\Gamma_1$ by a $\pi/4$ rotation. |
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