# Walking on a 3-sphere

1. Jan 8, 2016

### 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?

2. Jan 11, 2016

### Ben Niehoff

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

3. Jan 24, 2016

### 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.