The solution to Einstein's equations for a three-dimensional spacetime with a single point mass looks like a flat spacetime everywhere to local observers -- except when they make measurements around spatially closed curves (curves that come back to the same place, but at a later time) that contain the mass itself.
A Flatlander travelling around the blue arc will think she's been around a complete circle. But she will measure the circumference of this circle to be smaller than 2 R Pi, or 6 Pi in this case. The angular size of the missing wedge above is Pi/2, so the Flatlander will measure the circumference of her path to be 3 R Pi/2, or 4.5 Pi. Therefore she will be able to deduce that her path has encircled a point mass, and that she must live on a cone, not on a flat plane. Note that circular paths that do not circle the mass will have the normal circumference of 2 R Pi