Flat tori in $S^3$ arising as Hopf preimages of closed curves on $S^2$, projected stereographically to $\mathbb{R}^3
If γ:S1→S2 is a simple closed curve on the sphere, its preimage under the Hopf map π:S3→S2 is topologically a torus that inherits a flat metric from the round 3-sphere. A theorem of Ulrich Pinkall shows that all flat tori arise this way: if L is the curve’s length and A the enclosed area, the resulting torus is isometric to the quotient of R2 by the lattice generated by (2π,0) and (A/2,L/2).
This program provides an interactive graphing calculator depicting preimages of curves that oscillate sinusoidally about the equator of S2, projected into R3 via stereographic projection. Adjusting the amplitude of oscillation changes the conformal class of the resulting flat torus.
Controls: Click and drag to orbit, scroll to zoom.
