Walk a 3-sphere w/Hopf Fibration: Parameterization

[Hornbein]

Consider a three-sphere with a Hopf fibration. Choose a point on one fiber. Move an infinitesimal distance ds perpendicular to that fiber to reach a point on another fiber. Repeat ad infinitum. What is a parameterization of the resulting path?

 

[Ben Niehoff]

You would end up walking in a great circle. Parametrize that however you like.

 

[zinq]

All distances are either positive or zero; there are no infinitesimal distances.

But if you are asking about a differentiable path on the 3-sphere S³ whose tangent vectors are always perpendicular to the fibres of the Hopf fibration . . .

(and note that the Hopf fibration may be thought of as a specific mapping

p: S³ → S²​

from the 3-sphere to the 2-sphere)

. . . then your path is equivalent to specifying a starting point x on S³, and also a path on S², beginning at p(x), that represents the path on S³ after it has been pushed down to S² by applying the mapping p to each point of your path.

In other words, given the starting point x on S³, and the image under p of your path in S², that is all you need to know in order to uniquely reconstruct your path on S³.