differential geometry: smooth atlas of an ellipsoid


by gotmilk04
Tags: atlas, diff geometry, geometry, manifold, smooth
gotmilk04
gotmilk04 is offline
#1
Sep23-11, 10:12 AM
P: 45
1. The problem statement, all variables and given/known data

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.

2. Relevant equations


3. 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?
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