how can I transform my calculation on S^3 to the S^2.
for example a trace or a Fourier transform
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.
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.
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.
May be Maple atlas package can help to make some real calculations.
See http://www.mathshop.digi-area.com/prod/atlas/index.php [Broken]
It can make calculations for manifolds and mapping one into another.
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