Geometry of d=3 with one point mass, according to Einstein equations

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. When they do this, they learn that the space part of the spacetime they live in is not a flat plane, but a cone. 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 The missing angle is called the deficit angle. According to Einstein's equations, the mass M of the point is related to the deficit angle B through the formula B=8GM Pi, where G is Newton's constant. The deficit angle can't be larger than 2Pi, so there is a limit on the allowed mass: M must be less than or equal to 1/4G. (Note: the units of Newton's constant are not the same in three and four spacetime dimensions. Why not?) Another way of picturing this cone is shown below. It's just the xy plane with a wedge removed, and the sides sewn back together. Note that the blue circular arc is actually a closed path when the red dashed lines are identified by the sewing-together operation.