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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?

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    $\begingroup$ Yes, this is a theorem of Moser: see "On the volume elements of a manifold", jstor.org/stable/1994022. $\endgroup$
    – macbeth
    Commented Dec 3, 2011 at 20:56

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Here is an "answer-version" of my comment:

Yes, this is true in general. The reference I know is Moser's 1965 paper "On the volume elements on a manifold" (http://www.jstor.org/stable/1994022).

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 $\varphi^*\tau=\sigma$.

The orientability hypothesis isn't really necessary (just use densities rather than volume forms; see Moser's footnote (2)).

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