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Walking on a 3-sphere

  1. Jan 8, 2016 #1
    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?
     
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  3. Jan 11, 2016 #2

    Ben Niehoff

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    You would end up walking in a great circle. Parametrize that however you like.
     
  4. Jan 24, 2016 #3
    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.
     
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