Let $(S,g)$ be a Riemannian surface. Then the curvature function $\kappa: S\to \mathbb{R}$ can be counted as a random variable. So it produce a probability density function $f_g:\mathbb{R}\to \mathbb{R}$. Similarly every Riemannian manifold $(M,g)$ introduces a probability density function $f_g$ associated to the scalar curvature $R:M\to \mathbb{R}$ as a random variable.

Question: Can one equipe a compact surface $S$ with a Riemannian metric $g$ such that the resulting probability distribution would be a uniform distribution, i.e the density function $f_g$ would be a constant function? What about the case of Riemannian manifolds, namely dimension greater than 2? If the answer is yes can we choose an analytic Riemannian metric with the above mentioned property? (The distribution associated to the curvature function would be uniform.)