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?