Coplanarity of lines in projective and polar Grassmann spaces

Abstract

Abstract. The structure of lines in a projective or a polar Grassmann space together with binary coplanarity is a sufficient system of primitive notions for these geometries.Mathematics Subject Classification. 51A45, 51A50.

Geometry on the lines of spine spaces

“…In the current paper spine spaces are considered in view of coplanarity relation. The same problem was discussed in [8] for general Grassmann spaces, i.e. Grassmann spaces and polar Grassmann spaces.…”

“…Both spines spaces and polar Grassmann spaces are fragments of Grassmann spaces. It has to be noted here that the proof of Lemma 1.3 in [8] fails to be complete. In this paper, apparently in the context of spine space, we prove that lemma as Lemma 2.2.…”

“…In spine spaces 3 lines satisfying ∆ δ 3 determine a δ-clique. From this point of view, ∆ δ n could be defined as a ternary relation in (8). The reason why we introduce so general formula will be explained later in Erratum A.…”

“…It was proved in [8] that the structure of lines together with binary coplanarity relation π is a sufficient system of primitive notions for projective and polar Grassmann spaces. Following the ideas, motivations, and objectives of that paper we use similar methods to prove the same for spine spaces.…”

“…Following the ideas, motivations, and objectives of that paper we use similar methods to prove the same for spine spaces. This is also the reason why our terminology and notation is heavily based on [8]. In consequence many definitions and concepts may sound familiar for the reader.…”