{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW06 - Differential Geometry 1 Homework 06 Let α ∈...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Differential Geometry 1 Homework 06 Let α ∈ Ω1 (M ) be nowhere-zero 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 well-defined cohomology class 3 [γ ∧ dγ ] ∈ HdR (M ). 1 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online