The map \mathbb{C}^2 \rightarrow \mathbb{CP}^1 = S^2  restricted to S^3 \subset \mathbb C^2 is called the Hopf map and is usually viewed as a map (fibration) S^3 \rightarrow S^2 whose fibers (preimage of a point) are circles.

The preimage of a circle in S^2 is called a Clifford torus. These fill out all of S^3 except for two linked circles, and are in turn foliated by circles.

To visualize the Hopf fibration, we apply stereographic projection S^3 \rightarrow \mathbb{R}^3 or, better yet, into the Heisenberg group. Here's a picture of three Clifford tori with Heisenberg horisontal lines:

Hopf Fibration with Heisenberg Horisontal Lines

 

Hopf Fibration with Heisenberg Horisontal Lines


 Each Clifford torus lies along the XY plane. Intersected with this plane, it becomes two circles, of inverse radii.

 The horizontal lines on the Clifford tori either close up or become dense, depending on the rationality of the radius (or perhaps of a related number, I'm not sure). Below, I take a Clifford torus and draw horisontal lines through five equally spaced points, and then very slowly increase the radius of the torus, redrawing the picture with each adjustment.

In case you're interested in seeing the circles of the Hopf fibration, I recommend looking at Paul Nylander's website or the Wikipedia article on the Hopf fibration.

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Last Updated ( Thursday, 07 May 2009 22:47 )