Inspired by a nice recent MO question, I thought I would ask a similar one: which projective planes $P$ can be given a topology and structure of a smooth connected manifold where the lines form smooth submanifolds (and maybe I need to require they intersect transversely?)?
Note - the connectedness hypothesis rules out just taking a discrete topology - maybe replacing "connected" with "positive dimensional" yields the same answer?
Here I am thinking of synthetic projective planes - so $P$ is a set (the set of points) together with a collection of subsets (the lines) satisfying some axioms (two points lie on a unique line, every pair of distinct lines intersect in a unique point, and maybe some non-degeneracy conditions like the existence of 3 non-collinear points and every line containing at least 3 points).
