I'm wondering if the notion of an orientable/non-orientable manifold has any reasonable extension that allows for a similar classification of finite geometries.


For example, the real projective plane is non-orientable. Does this somehow mean that a finite projective plane is "non-orientable" too?


I would also be interested in a definition that applies over a narrower domain, e.g. a notion of orientability for finite *ordered* geometries or something like that.



Is there a well-known notion of orientability for finite geometries?
To do that you would need a notion of a non-orientable and an orientable linear transformation, i.e., essentially a notion of a "positive" and "negative" determinant, where "positive" determinants would form a subgroup *not* containing the element $-1$. This works for the field $\mathbb{Z}/p\mathbb{Z}$ if and only if the prime $p$ satisfies $p\equiv 3 (mod\; 4)$.


Thus denoting by $G\subset (\mathbb{Z}/p\mathbb{Z})^\ast$ the index-2 subgroup consisting of the quadratic residues modulo $p$, one obtains an "orientable double cover" $M=(F^3\setminus\{0\})/G$, where $F=\mathbb{Z}/p\mathbb{Z}$. Quotienting $M$ by the ("orientation-reversing") antipodal map, you get the projective plane over $F$.


