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Prove that x^4+y^4=1 is a manifold

  1. Jun 13, 2009 #1
    Can someone help me out whit this
    I proved for the circle, but I can't prove it for this

    -Prove that x^4+y^4=1 (the set of points) is a manifold

    For the circle it was easy, but how do I take on this case?

    Thanks
     
    Last edited: Jun 14, 2009
  2. jcsd
  3. Jun 14, 2009 #2
    Re: diffeomorphism

    how have you tried to prove it?
     
  4. Jun 14, 2009 #3
    Re: diffeomorphism

    I was thinking about a kind of stereographic projection
    but I didn't know how to do this
     
  5. Jun 14, 2009 #4
    Re: diffeomorphism

    a kind of stereographic projection to create coordinate neighborhoods?
     
  6. Jun 14, 2009 #5
    Re: diffeomorphism

    yes exactly
    Do you think that this will work? Then I try it again
     
  7. Jun 14, 2009 #6
    Re: diffeomorphism

    my guess is that you should be able to identify open intervals that project back to give coordinate charts. This would give you a differentiable manifold. You then need to argue that it is diffeo to a circle.
     
  8. Jun 14, 2009 #7

    gel

    User Avatar

    Re: diffeomorphism

    Let f(x,y) be any smooth real valued function, and a be a real number. Then, the set of points (x,y) for which f(x,y) = a and df/dx, df/dy are not both zero is a manifold.

    In this case, f(x,y) = x^4 + y^4, and df/dx = 4x^3, df/dy=4y^3, which can't both be zero when f=1. So, yes, the locus of points you describe is a manifold.

    That's a very general method. In this case you can show that the locus of points is diffeomorphic to the circle, as already mentioned.
     
  9. Jun 14, 2009 #8
    Re: diffeomorphism

    right but the problem was to construct coordinate neighborhoods. An appeal to the implicit function theorem is not a direct construction. But there is more to it. Once you know it is a manifold you need to prove that it is a topological circle (without appealing to the structure theorem for 1 manifolds.) With coordinate charts this should be easy to do.
     
  10. Jun 14, 2009 #9

    gel

    User Avatar

    Re: diffeomorphism

    if A = {(x,y) : x^4+y^4=1} and S={(x,y):x^2+y^2=1} is the unit circle, then you can define

    [tex]
    \begin{align*}
    &f\colon A\to S,\ f(x,y) \equiv (\lambda x, \lambda y),\ \lambda = (x^2+y^2)^{-\frac{1}{2}}\\
    &g\colon S\to A,\ g(x,y) \equiv (\lambda x, \lambda y),\ \lambda = (x^4+y^4)^{-\frac{1}{4}}
    \end{align*}
    [/tex]

    which are smooth maps and inverse to each other, showing that A,S are diffeomorphic.
     
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