It is known that in flat Euclidean Space the internal angles of a triangle sum to $180^{\circ}$ but if we draw a similar triangle in curved space then the sum of interior angles will be more or less depending on whether the curvature is positive or negative, respectively.
Is it possible to construct a geometric figure in curved space such that in the large $N$ limit, where $N$ is the number of sides of the polygon, we have the following two conditions :
The angle between any two sides is same i.e. $\Delta \theta_{ij}\ \forall\ i,j$ representing two different sides of polygon.
The distance between any two (and not mere adjacent) vertices remains same?
I don't know how one would proceed to solve the problem or what kind of mathematical language would be used here or if this is a solvable problem.
Any insights would be much appreciated.