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Maifold calculus

  1. Nov 4, 2005 #1
    how can I transform my calculation on S^3 to the S^2.
    for example a trace or a Fourier transform
  2. jcsd
  3. Nov 6, 2005 #2


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    hopf map? consider R^4 as C^2, the complex 2 space, and consider all complex "lines" in C^2. this family of complex subspaces is homeomorphic to S^2, hence this fibres S^3 over S^2 with fibers equal to circles. this is the famous hopf map from S^3 to S^2.
  4. Nov 6, 2005 #3
    manifold calculus

    well my problem is: I have an integral in S^3 . I want to calculate this integral.
    i) in S^3.
    ii) how can I transform this integral to S^2.

    If you think that you need more explanation I would be glad to
    sent it for you.
  5. Nov 6, 2005 #4


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    you might try to generalize the fubini theorem, i.e. integrate over the fibering circles first and then integrate those integrals over the 2 sphere.

    but this is only indicated if the quantity being integrated somehow restects the compex circles in the hopf fibration.
  6. Nov 10, 2005 #5
    May be Maple atlas package can help to make some real calculations.
    See http://www.mathshop.digi-area.com/prod/atlas/index.php
    It can make calculations for manifolds and mapping one into another.
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