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HW06sol - Dierential Geometry 1 Homework 06 Let 1(M be...

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Differential Geometry 1 Homework 06 Let α Ω 1 ( M ) be nowhere-zero and choose ζ Vec M such that α ( ζ ) is identically one. 1. Show that if β Ω k +1 ( M ) and α β = 0 then β = α γ for some γ Ω k ( M ). [ Hint : Recall that ζ is an antiderivation.] 2. Assume that α d α = 0 and choose γ Ω 1 ( M ) such that d α = α γ . Show that γ d γ Ω 3 ( M ) is closed; show further that if also d α = α γ then γ d γ - γ d γ is exact. [ Hint : First show α d γ = 0.] Remark : This shows that α gives rise to a well-defined cohomology class [ γ d γ ] H 3 dR ( M ). Solutions (1) Very brief: If α β = 0 then 0 = ζ ( α β ) = ( ζ α ) β - α ( ζ β ) as is an antiderivation; as ζ α 1 it follows that β = α ( ζ β ). (2) From above, d α = α γ for some γ Ω 1 ( M ). Now 0 = d 2 α = d( α γ ) = d α γ - α d γ = - α d γ because d α γ = ( α γ ) γ = 0; from above again, it follows that d γ = α ε for some ε Ω 1 ( M ). To see that γ d γ is closed, compute: d( γ d γ ) = d γ d γ = ( α ε ) ( α ε ) = 0 . To see that γ d γ - γ d γ is exact, first let d

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HW06sol - Dierential Geometry 1 Homework 06 Let 1(M be...

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