Establishing a smooth differential structure on the ellipsoid

• saminator910
In summary, the conversation discusses the construction of a C∞ natural differential structure on the ellipsoid given by the equation (x1/a)^2 + (x2/b)^2 + (x3/c)^2 = 1. The question of whether this structure is diffeomorphic to S2 is raised, and the individual discusses their attempts at constructing charts and coordinate transformations to prove smoothness and a diffeomorphism. One chart uses a stereographic projection and a "drop the z coordinate" mapping, while another maps positive coordinates to S2. The individual is unsure if they are on the right track towards constructing the diffeomorphism."
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?

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1. What is an ellipsoid and why is it important to establish a smooth differential structure on it?

An ellipsoid is a three-dimensional geometric shape that resembles a stretched-out sphere. It is important to establish a smooth differential structure on it because it allows us to define a coordinate system and perform mathematical calculations, making it easier to analyze and understand the properties of the ellipsoid.

2. How is a smooth differential structure established on an ellipsoid?

In order to establish a smooth differential structure on an ellipsoid, a set of smooth coordinate charts must be defined on the surface. These charts map points on the ellipsoid to points in a Euclidean space, allowing us to perform mathematical operations and define vectors and tensors on the surface.

3. What are some challenges in establishing a smooth differential structure on an ellipsoid?

One challenge is ensuring that the coordinate charts cover the entire surface of the ellipsoid without any gaps or overlaps. Another challenge is finding a way to smoothly transition between different coordinate charts at the boundaries.

4. How is the smooth differential structure on an ellipsoid related to its curvature?

The smooth differential structure on an ellipsoid is closely related to its curvature. In fact, the smoothness of the coordinate charts and their transition functions determine the smoothness of the curvature on the surface.

5. What are some real-world applications of establishing a smooth differential structure on an ellipsoid?

Establishing a smooth differential structure on an ellipsoid has many practical applications, such as in geodesy for accurately measuring the Earth's shape and in geophysics for understanding the Earth's internal structure. It is also used in computer graphics for creating realistic 3D models of ellipsoidal objects.

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