# How to distinguish between walking on a sphere and walking on a projective plane?

Inspired by the similar question regarding a torus, imagine that you're a flatlander walking in your world. How could you distinguish between your world being a sphere versus being a projective plane?

It seems like this would be somewhat harder because you can't necessarily use an argument about non-positive curvature, and you can't (easily?) take advantage of the fact that the projective plane is non-orientable and not embeddedable in $\mathbb{R}^3$.

I would also be interested in any methods that could be used in this case, would wouldn't tell you anything substantial if you were on a torus.

• Unless you can define precisely what "walking on a surface" means (and as you can see in the comments below this is indeed an issue), then this is a soft question. – Najib Idrissi Nov 3 '16 at 10:06

Since the projective plane is $\mathbb{Z}_2$ quotient of the sphere, it is impossible to distinguish the two if you confine yourself to a simply-connected region.
• @MikePierce, more precisely, if you say that "you would have to walk over that same loop twice...", given that the fundamental group of projective plane is $\mathbb{Z}_2$, you are literally saying that we live on the universal covering of the projective plane, which is the sphere. – Peter Kravchuk Nov 3 '16 at 7:05