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# HW05sol - Dierential Geometry 1 Homework 05 1 Let(x y z be...

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Differential Geometry 1 Homework 05 1. Let ( x, y, z ) be the standard coordinates on R 3 and ( X, Y ) those on R 2 . Let ω Ω 2 ( S 2 ) be the standard two-form: ω = x d y d z + y d z d x + z d x d y and let F : R 2 S 2 be inverse to stereographic projection from the north pole (0 , 0 , 1). Prove that F * ω = - 4 ( X 2 + Y 2 + 1) 2 d X d Y. 2. Prove explicitly that the two-sphere has trivial de Rham cohomology in degree one: H 1 dR ( S 2 ) = 0 . Suggestion : Consider stereographic projection from the N and S poles. Solutions (1) In terms of the coordinates as given, note that x F = 2 X R 2 + 1 , y F = 2 Y R 2 + 1 , z F = R 2 - 1 R 2 + 1 where R 2 = X 2 + Y 2 . It follows that F * (d x ) = d( x F ) = 2 ( R 2 + 1) 2 (1 + Y 2 - X 2 )d X - 2 XY d Y F * (d y ) = d( y F ) = 2 ( R 2 + 1) 2 (1 + X 2 - Y 2 )d Y - 2 Y X d X F * (d z ) = d( z F ) = 4 ( R 2 + 1) 2 X d X + Y d Y . Writing S = R 2 + 1 for further convenience, it follows that F * ( x d y d z ) = 16 S 5 X - 2 XY 2 - (1 + X 2 - Y 2 ) X d X d Y 1

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or F * ( x d y d z ) = - 16 X 2 S 4 d X d Y and likewise F * ( y d z d x ) = - 16 Y 2 S 4 d X d Y while F * ( z d x d y ) =
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HW05sol - Dierential Geometry 1 Homework 05 1 Let(x y z be...

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