HW05sol - Differential Geometry 1 Homework 05 1. Let (...

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Unformatted text preview: 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 2 ( S 2 ) be the standard two-form: = 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 , , 1). Prove that F * =- 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 ....
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HW05sol - Differential Geometry 1 Homework 05 1. Let (...

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