Which mathematical constant relates a circle's circumference to its diameter in any curved space?

None does - pi is the answer only in flat space. Draw a circle on a sphere and its circumference is less than pi times its diameter, because the diameter measured across the curved surface is longer than the flat geometry expects. On a saddle-shaped surface the ratio is greater.

That gives a way to measure the shape of a space from inside it, with no outside view. Gauss realised a surface's curvature is intrinsic - detectable by measurements made entirely within it - and reportedly measured the angles of a triangle between three mountain peaks to see whether they came to 180 degrees. This is what lets cosmologists ask whether the universe is curved without standing outside it; the current answer, from the cosmic microwave background, is flat to within the measurement error.

The stubbornly persistent story is that the Indiana legislature once tried to define pi as 3. A bill in 1897 did contain a construction implying several wrong values, and it passed the House; a mathematician who happened to be in the building on other business explained the problem to the senators, who postponed it indefinitely. Nobody legislated a value, and the bill was about a claimed squaring of the circle rather than about pi as such.