differential geometry: smooth atlas of an ellipsoidby gotmilk04 Tags: atlas, diff geometry, geometry, manifold, smooth 

#1
Sep2311, 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: LS[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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