- #1
cianfa72
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- TL;DR
- About the Hopf fibration of the 3-sphere
As shown by this animation, the fibers of the Hopf fibration of the 3-sphere are circles (click on a point on the sphere to visualize the associated fiber). As far as I understand, they never intersect and their union is the 3-sphere itself.
I'd be sure whether the circles in the animation are given by stereographic projection of the 3-sphere from a point, say the "equivalent" of the ##S^2## north-pole. Assuming the viewpoint of 3-sphere defined by its embedding in ##\mathbb C^2## as "ambient" space, the 3-sphere is given as the locus of complex pairs ##(z,w)## such that ##|z|^2 + |w|^2 = 1##. Then one can employ, for instance, the stereographic projection from the complex point ##(0,i)## that is ##(0,0,0,1)## using the natural identification ##\mathbb C^2 \cong \mathbb R^4##. Such a stereographic projection maps the 3-sphere on the 3D space ##v=0## where ##z = x +iy, w = u + iv## including ##\infty##.
Did I understand it correctly ? Thanks.
I'd be sure whether the circles in the animation are given by stereographic projection of the 3-sphere from a point, say the "equivalent" of the ##S^2## north-pole. Assuming the viewpoint of 3-sphere defined by its embedding in ##\mathbb C^2## as "ambient" space, the 3-sphere is given as the locus of complex pairs ##(z,w)## such that ##|z|^2 + |w|^2 = 1##. Then one can employ, for instance, the stereographic projection from the complex point ##(0,i)## that is ##(0,0,0,1)## using the natural identification ##\mathbb C^2 \cong \mathbb R^4##. Such a stereographic projection maps the 3-sphere on the 3D space ##v=0## where ##z = x +iy, w = u + iv## including ##\infty##.
Did I understand it correctly ? Thanks.
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