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?

  • 9
    $\begingroup$ Yes, this is a theorem of Moser: see "On the volume elements of a manifold", jstor.org/stable/1994022. $\endgroup$
    – macbeth
    Dec 3, 2011 at 20:56

1 Answer 1


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)).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.