Geometry of the parallelism in polar spine spaces and their line reducts

Abstract

The concept of spine geometry over a polar Grassmann space was introduced in [9]. The geometry in question belongs also to a wide family of partial affine line spaces. It is known that such a geometry -- e.g. the ``ordinary'' spine geometry, as considered in [13, 14] can be developed in terms of points, so called affine lines, and their parallelism (in this case this parallelism is not intrinsically definable: it is not `Veblenian', cf. [11]). This paper aims to prove an analogous result for polar spine spaces. As a by-product we  obtain several other results on primitive notions for the geometry of polar spine spaces.