# On a compact manifold, what kind of function can be the Jacobian of a diffeomorphism?

I could not answer or find references of this question, even for the following special case:

On $S^2$ (the two-sphere equiped with the standard Riemannian metric), is every positive smooth function with integral $1$ the Jacobian of some diffeomorphism?

An equivalent formulation of the question is: On $S^2$, is every positive smooth probability measure the translate of the standard one by some diffeomorphism?

-
Yes, this is a theorem of Moser: see "On the volume elements of a manifold", jstor.org/stable/1994022. –  macbeth Dec 3 '11 at 20:56
@macbeth: you should make your comment an answer. If you're determined not to get credit for your answer, you could always tag your answer "community wiki" when you create it. –  Ryan Budney Dec 3 '11 at 22:01
Specifically, let $M$ be a compact connected orientable manifold, and let $\sigma$ and $\tau$ be smooth volume forms on $M$ both with integral 1. Then there exists a diffeomorphism $\varphi:M\to M$ such that $\varphi^*\tau=\sigma$.