Suppose you are transported to an 2 dimensional hyperbolic world, ( a plane (2 dinensional) manifold with a constant negative curvature ) the only geometrical tools you have are a ruler, a pencil, a measuring stick (but with inches , not the absolute hyperbolic scale) a piece of string and an right angle,
How to measure the gaussian / total curvature of where you have landed?
(or "How many inches go into an absolute hyperbolic distance of 1 ?")
Limitations
- you can use the piece of string to make circles ,
- you cannot use the piece of string and the ruler to measure curves. (just not
The Absolute hyperbolic distance is a measurement in hyperbolic geometry, if trilateral ABC has angles (measured in radians) $\angle ABC = \frac{\pi}{2}$ (right angle) , $\angle BAC = 1$ and $\angle ACB = 0$ ( $C$ is an Ideal point) then $AB$ has the absolute distance 1
I don't know what you mean by "absolute hyperbolic scale" (perhaps you're referring to some model of hyperbolic space embedded in $\mathbb{R}^n$?), but distance is distance---part of the point of abstract Riemannian manifolds is that your metric (ruler) is part of the data of your space, and there are no other metrics sitting around.
Anyway, as for your question, one easy option would be to use the following characterization of scalar curvature: The circumference of a circle of radius $r$ on a Riemannian surface at a point with scalar curvature $\kappa$ satisfies $$C = 2 \pi\left(1 - \frac{S}{12} r^2 + O(r^4)\right).$$
So, measure off some small length $r$ of the string using the ruler, tie the string around the pencil and use it as a makeshift compass to draw a small circle of radius $r$ around any point, and use the rearranged formula $$S = \frac{1}{r^2} \left(12 - \frac{6C}{\pi} \right) + O(r^2).$$ The error term will be small when $r \ll |\kappa|$.