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Parsing the Fibers of Critical Values

  1. Jul 23, 2012 #1

    WWGD

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    Hi, All:

    I am trying to better understand what happens at fibers of critical values:

    Specifically, I had the fibers of the map:

    f(z1,z2): ℂ2→ℂ , given by:

    f(Z1, Z2)= Z12+z22 =0

    I rewrote it as a map from ℝ4→ℝ2=

    f(x1,y1,x2,x2)=

    (x12-y12+x22-

    y22 , x1y1+2x2y2)


    Which has the Jacobian:
    (Sorry, I don't know how to Tex a matrix ):

    [ 2x1, 2y1, 2x2, 2y2 ]

    [ 2y1, 2x1, 2y2, 2x2





    Now, by one of the family of theorems in { inverse function, implicit function, regular rank theorem} , the fibers of the non-zero values are submanifolds of ℂ2.

    After row-reducing, we get that the only critical value is (0,0) .

    Now, some work on f{-1}(0,0):

    i)Not very helpful: Z12+Z22=0

    implies Z1=iZ2.

    ii) A little better:

    Restricting to (x1,x2, 0, 0):

    Then the general fiber is a circle, and the collection of all fibers is a cone .

    Is there anything else we can do to get a better understanding of what the general

    fibers are like ? I think all the regular fibers are isomorphic to each other--tho I cannot

    find a good proof -- and then we have the critical fiber.

    Any ideas?

    Thanks.
     
    Last edited: Jul 23, 2012
  2. jcsd
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