This is one idea that got me thinking.

In a space with positive curvature, we can have two basic geometries: spherical, where two straight lines intersect in two points, and elliptic, where they only intersect in one point.

But could this be extended higher? Would it be possible to have a self-consistent geometry of, say, double-winded sphere, where two straight lines intersect in four points and the total area of the world is double the area of a normal sphere?