Unformatted text preview: Diﬀerential Geometry 1
Homework 06 Let α ∈ Ω1 (M ) be nowherezero and choose ζ ∈ VecM such that α(ζ ) is
identically one.
1. Show that if β ∈ Ωk+1 (M ) and α ∧ β = 0 then β = α ∧ γ for some
γ ∈ Ωk (M ). [Hint: Recall that ζ is an antiderivation.]
2. Assume that α ∧ dα = 0 and choose γ ∈ Ω1 (M ) such that dα = α ∧ γ .
Show that γ ∧ dγ ∈ Ω3 (M ) is closed; show further that if also dα = α ∧ γ
then γ ∧ dγ − γ ∧ dγ is exact. [Hint: First show α ∧ dγ = 0.]
Remark: This shows that α gives rise to a welldeﬁned cohomology class
3
[γ ∧ dγ ] ∈ HdR (M ). 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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