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Establishing a smooth differential structure on the ellipsoid

  1. Jan 4, 2013 #1
    1. The problem statement, all variables and given/known data
    Construct a C natural differential structure on the ellipsoid

    [itex]\left\{(x_{1}, x_{2}, x_{3})\in E | \frac{x_{1}^{2}}{a^{2}}+\frac{x_{2}^{2}}{b^{2}}+ \frac{x_{3}^{2}}{c^{2}}=1\right\}[/itex]

    Is this diffeomorphic to S2? Explain.


    2. Relevant equations

    Do I need to prove homeomorphism for my functions mapping E to ℝ2?
    How to/ do I need to prove smoothness for my coordinate transformations, and my diffeomorphism to S2? Are my charts valid?, I use one stereographic projection chart for the ellipsoid minus 1 point, then a "drop the z coordinate" mapping for the top half including the point I missed.

    3. The attempt at a solution
    Here are my charts,

    [itex]\varphi:U\rightarrowℝ^{2}[/itex]
    [itex]U = E-{(0,0,c)}[/itex]
    [itex]φ(x_{1},x_{2},x_{3}) = (\frac{x_{1}}{c-x_{3}},\frac{x_{2}}{c-x_{3}},0)[/itex]

    [itex]ψ:V\rightarrowℝ^{2}[/itex]
    [itex]V = \left\{(x_{1}, x_{2}, x_{3})\in V | x_{3}>0\right\}[/itex]
    [itex]φ(x_{1},x_{2},x_{3}) = (x_{1},x_{2},0)[/itex]

    To be a differential structure, the coordinate transformation must be smooth
    [itex]φψ^{-1}:ψ(U\cap V)\rightarrowℝ^{2}[/itex]
    [itex]φψ^{-1}(x_{1},x_{2},x_{3})=(\frac{x_{1}}{c-\sqrt{1-x_{1}^{2}-x_{2}^{2}}},\frac{x_{2}}{c-\sqrt{1-x_{1}^{2}-x_{2}^{2}}},0)[/itex]
    It is pretty clear to me these charts are smooth for the values it would need to operate on, need to prove?

    This is where it gets dicey, I need to find a smooth mapping from the ellipsoid to the 2 sphere, will I need multiple charts, here is one for the positive coordinates.
    [itex]F = \left\{(x_{1}, x_{2}, x_{3})\in F | x_{3}>0\right\}[/itex]
    [itex]f(x_{1},x_{2},x_{3}) = (a^{2}x_{1}^{2}, b^{2}x_{2}^{2}, c^{2}x_{3}^{2})[/itex]
    [itex]f(F) = \left\{(x_{1}, x_{2}, x_{3})\in S^{2} | x_{3}>0\right\}[/itex]

    So, am I on the right track to construct this diffeomorphism?
     
    Last edited: Jan 4, 2013
  2. jcsd
  3. Jan 5, 2013 #2
    Bump......
     
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