HW05 - Differential Geometry 1 Homework 05 1. Let (x, y, z...

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Unformatted text preview: Differential Geometry 1 Homework 05 1. Let (x, y, z ) be the standard coordinates on R3 and (X, Y ) those on R2 . Let ω ∈ Ω2 (S 2 ) be the standard two-form: ω = xdy ∧ dz + y dz ∧ dx + z dx ∧ dy and let F : R2 → S 2 be inverse to stereographic projection from the north pole (0, 0, 1). Prove that F ∗ω = − (X 2 4 dX ∧ dY. + Y 2 + 1)2 2. Prove explicitly that the two-sphere has trivial de Rham cohomology in degree one: 1 HdR (S 2 ) = 0. Suggestion: Consider stereographic projection from the N and S poles. 1 ...
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