Negative curvature of space-time: the col on Hesk Fell

In a most amazing bit of outdoor Physics, this col between Stainton Pike and Hesk Fell above Devoke Water was used to describe negatively curved space to me in a way that I'd not heard before. The col is like a saddle. In the picture, it goes down and then up away from us onto Hesk Fell. Left to right it curves the opposite way. Now imagine banging in a peg in the middle with a rope of a certain length attached to it. You are on the end of the rope and walk round in a circle. On the flat, the circumference comes out to be 2 x pi x radius. On this surface, it is bigger than 2 x pi x radius. That means that the surface has negative curvature. Why does it happen? Because the twisting of the land stretches the space that makes up the surface so there is more space to cover. I understood this forum post http://math.stackexchange.com/questions/619063/circumference-of-a-circle-in-hyperbolic-space but I'd like to get to grips with the maths. Apparently it is a hyperbolic space and Roger Penrose does write about that!