## differential geometry: smooth atlas of an ellipsoid

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

Consider the ellipsoid L $\subset$E3 specified by

(x/a)^2 + (y/b)^2 + (z/c)^2=1

(a, b, c $\neq$ 0). Define f: L-S$^{2}$ by f(x, y, z) = (x/a, y/b. z/c).

(a) Verify that f is invertible (by finding its inverse).
(b) Use the map f, together with a smooth atlas of S$^{2}$, to construct a smooth atlas of L.

2. Relevant equations

3. The attempt at a solution
For part (a), would the inverse be f$^{-1}$(x/a, y/b, z/c)= (x,y,z)?
So that you take the points on the ellipsoid and get points on S$^{2}$?

For (b), a smooth atlas of S$^{2}$ is
U$_{1}$= {(x,y,z)$\in$S$^{2}$|(x,y,z)$\neq$(1,0,0)}
U$_{2}$= {(x,y,z)$\in$S$^{2}$|(x,y,z)$\neq$(-1,0,0)}

But how do I use that with f to form an atlas of L?
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 Tags atlas, diff geometry, geometry, manifold, smooth

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