When is a linear map of 1-forms a pullback?

Asked Jan 23 '15 at 01:29

Active Feb 09 '15 at 16:04

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Every diffeomorphism $\phi: M\to N$ between two-dimensional compact oriented Riemannian manifolds induces a linear map on one-forms $L:\Omega^1(M)\to\Omega^1(N)$ given by the pullback of $\phi^{-1}$.

Is there a simple condition for when a linear map $L$ on the one-forms is the pullback by some diffeomorphism?

In principle, if I know how $L$ acts on exact forms, I know how any such $\phi$ would have to act on the zero-forms of $M$; I could then choose appropriate bases for $\Omega^0(M)$ and $\Omega^0(N)$ and read off which points of $M$ must map to which on $N$. And then check that this $\phi^{-1}$ correctly pulls back all of the coexact forms...

edited Jan 23 '15 at 02:55

asked Jan 23 '15 at 01:29

user7530

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What is a "coexact form" in this context? – Jesse Madnick Jan 23 '15 at 01:36
@JesseMadnick one-forms $*df$ where $f$ is a zero-form – user7530 Jan 23 '15 at 02:20
Ah, OK, I initially thought that $M$ and $N$ were just smooth manifolds; I see now that they are oriented Riemannian 2-manifolds. – Jesse Madnick Jan 23 '15 at 02:41
Yes, let me add oriented and compact to the question. – user7530 Jan 23 '15 at 02:55
Do you know of an example of an $L$ for which no $\phi^{-1}$ exists? – Jason DeVito - on hiatus Feb 09 '15 at 17:20
@JasonDeVito Consider for instance $N=M$ and an $L$ which acts as the identity on exact forms and negates coexact forms. – user7530 Feb 09 '15 at 18:52

When is a linear map of 1-forms a pullback?

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