Establishing a smooth differential structure on the ellipsoid

[saminator910]

Homework Statement

Construct a C^∞ natural differential structure on the ellipsoid

\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\}

Is this diffeomorphic to S²? Explain.

Homework 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.

The Attempt at a Solution

Here are my charts,

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

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

To be a differential structure, the coordinate transformation must be smooth
φψ^{-1}:ψ(U\cap V)\rightarrowℝ^{2}
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)
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.
F = \left\{(x_{1}, x_{2}, x_{3})\in F | x_{3}>0\right\}
f(x_{1},x_{2},x_{3}) = (a^{2}x_{1}^{2}, b^{2}x_{2}^{2}, c^{2}x_{3}^{2})
f(F) = \left\{(x_{1}, x_{2}, x_{3})\in S^{2} | x_{3}>0\right\}

So, am I on the right track to construct this diffeomorphism?

 

[saminator910]

Bump...