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Differential geometry: smooth atlas of an ellipsoid

  1. Sep 23, 2011 #1
    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?
     
  2. jcsd
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