Locally projective spaces which satisfy the Bundle Theorem

On projective spaces PG(r, q) with r ≥ 4

“…87, 2007 On projective spaces P G(r, q) with r ≥ 4 5 3 [2], [4], [5], [7], [8]. Thus, from now on we may suppose that n > q.…”

Linear spaces with many projective planes

“…In many papers the question is discussed which linear space (M, M) can be embedded into a projective space (cf. [1,2,3,4,5,9,10,12]). The usual method for projective embedding is the construction of the point set P of the projective space (P , L) by line bundles of the given linear space (M, M).…”

“…The crucial problem is to prove that every plane of (P , L) is a projective plane. If for example (M, M) is locally projective and satisfies the Bundle Theorem, then (P , L) is a projective space (cf [4,9]). For this proof one shows in a first step that every plane of P with a non-empty intersection with M is a projective space and in a second step that any plane of P is a projective plane.…”

Coplanarity of lines in projective and polar Grassmann spaces

“…Most of the time, e.g. in [1,15,18,34], the Bundle Theorem is the necessary condition for the construction of the bundle space in locally projective spaces. We do not follow exactly this path.…”

“…Theorem 1.1). We try to imitate the approach of [18] and other papers, e.g. [1,15,34], where the construction of a bundle space is used to recover various geometries embeddable into a projective space.…”