A diffeomorphism between manifolds (or surfaces) that preserves the mean value of functions Let $M$ and $N$ be two Riemannian manifolds with $f:N \to M$ a diffeomorphism with the following properties: for all  $u \in H^1(M)$, $\hat u := u\circ f$ satisfies $\hat u\in H^1(N)$ and furthermore the mean value of $u$ agrees with the mean value of $\hat u$: 
$$\frac{1}{|M|}\int_M u = \frac{1}{|N|}\int_N \hat u.$$
Is there some established term for such a diffeomorphism or such diffeomorphic surfaces (surfaces that preserve mean value)? Is such a condition really restrictive, as in, does it constrain the surfaces $M$ and $N$ can be too much? (An example where such condition holds can be two spheres $M$ and $N$ of different radius).

After applying a change of variables to the integral on $M$, the given condition is that $$\frac1{|M|}\int_N |\det Df| u \circ f = \frac1{|N|}\int_Nu\circ f$$
for every $u$. Thus it seems this condition restricts you to exactly the diffeomorphisms with constant Jacobian determinant. When $|N|=|M|$ these are known as measure-preserving diffeomorphisms, which are probably worth googling to find out more about - the behaviour should be essentially the same as in the general volume case, since you can just rescale.
