- #1

- 10

- 0

Hint: if dw=0 and [tex]f^{\star}w = d\eta, [/tex] consider [tex](f^{-1})^{\star}\eta. [/tex]

---------------------------------------------------

I'm not quite sure what the hint means or how to use it. is it true that [tex]f^{\star} [/tex] is basically another way of writing a differential - eg. df? I didn't really get a clear definition of it in my text.

anyway here's my thoughts so far:

consider [tex]w[/tex] as a form on U. suppose [tex]w[/tex] is closed. then dw = 0. since every closed form on U is exact, then there exists a [tex] \eta [/tex] on U such that [tex]w = d\eta [/tex].

but how is it that [tex] f^{\star}w = d\eta [/tex] (given in the hint)? like, how is this relationship derived? if [tex]w = d\eta [/tex] and also [tex] f^{\star}w = d\eta [/tex], then we have [tex]w = f^{\star}w [/tex]? I find that really confusing, and I'm not sure how to continue the problem. Any help is greatly appreciated. thanks in advance!