Geometries with bi-affine and bi-linear diagrams

We consider geometries belonging to the following diagram of rank n ≥ 4,. {A figure is presented}. We prove that when n ≥ 5, the only simply connected examples for this diagram arise from PG(n,q) by removing a hyperplane and the star of a point. We call these geometries bi-affine geometries. They are of two types, according to whether the point and the hyperplane chosen are incident or not. We also prove that there are just three types of flag-transitive simply connected examples for the rank 4 case of the above diagram, namely the two bi-affine geometries of rank 3 and the (well-known) two-sided extension of PG(2,4) for HS.