Other curvatures' π The ratio $\frac{\text{perimeter}}{\text{diameter}}$ of circles on the surface of constant curvature $0$ is constantly $3.14\dots$ and is called $\pi$. 


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*Is this ratio a constant for every other surface of constant curvature $\kappa$?

*How then (by which formula) does this constant depend on  $\kappa$?

The ratio is not constant on surfaces of non-zero curvature. Let $d(x)$ be the ratio of the perimeter of a circle of diameter $x$ by the diameter. If the curvature is negative then this is a strictly increasing function while if the curvature is positive it is a strictly decreasing function. It's quite easy to see that with particular models for such constant curvature spaces and directly computing. For instance, for circles on a sphere this is quite straightforward: the perimeter of the circle, as the diameter increases, reaches a maximum and then starts shrinking. 
I don't know of an exact formula, so will be looking forward to other answers. 
