Does a geometry exist where the circumference can be the square of the radius, $C = R^2$?

Question

In normal, Euclidean space, the circumference of a circle is a linear function of the radius: $$f(R) = 2 \pi R$$Does a geometry exist where the locus of points equidistant from a common point, $P_0$, such that the the distance around the surface (e.g. circumference in Euclidean space) is a function of the square of the radius (that is, the distance from $P_0$ to the surface) such that:$$f(R)=R^2$$ Is there anything to prevent me from hypothesizing such a shape?

Answer 1

If I understand correctly, the answer is no in dimension $2$, at least. By Bertrand-Diquet-Puiseux, the Gaussian curvature at the center of the circle would have to be $+\infty$. If I recall correctly, there are analogues of this formula for higher dimensions using the Ricci curvature, but I don't have any reference for it now.