Differential geometry: smooth atlas of an ellipsoid

[gotmilk04]

Homework Statement

Consider the ellipsoid L [itex]\subset[/itex]E3 specified by

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

(a, b, c [itex]\neq[/itex] 0). Define f: L-S[itex]^{2}[/itex] 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[itex]^{2}[/itex], to construct a smooth atlas of L.

Homework Equations

The Attempt at a Solution

For part (a), would the inverse be f[itex]^{-1}[/itex](x/a, y/b, z/c)= (x,y,z)?
So that you take the points on the ellipsoid and get points on S[itex]^{2}[/itex]?

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

But how do I use that with f to form an atlas of L?

 

[mighty2000]

To construct a smooth atlas of L, we can use the inverse map of f to map the open sets U_1 and U_2 onto L. Let's call these sets V_1 and V_2 respectively. Then, we can define the following smooth charts:

\phi_1: V_1 \rightarrow U_1, \quad \phi_1(x/a, y/b, z/c) = (x,y,z)

\phi_2: V_2 \rightarrow U_2, \quad \phi_2(x/a, y/b, z/c) = (x,y,z)

These charts are smooth because they are the composition of the smooth map f^{-1} and the smooth charts of S^2. Together, these charts form a smooth atlas of L.