# Establishing a smooth differential structure on the ellipsoid

1. Jan 4, 2013

### saminator910

1. The problem statement, all variables and given/known data
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.

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,

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

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

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?

Last edited: Jan 4, 2013
2. Jan 5, 2013

Bump......