- #1

saminator910

- 96

- 1

## Homework Statement

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 S

^{2}? Explain.

## Homework Equations

Do I need to prove homeomorphism for my functions mapping E to ℝ

^{2}?

How to/ do I need to prove smoothness for my coordinate transformations, and my diffeomorphism to S

^{2}? 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,

[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?

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