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 twoform:
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 twosphere 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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This note was uploaded on 07/08/2011 for the course MTG 6256 taught by Professor Robinson during the Spring '09 term at University of Florida.
 Spring '09
 Robinson

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