Coplanarity of lines in projective and polar Grassmann spaces (Q291296)

Let \((S,\mathcal L)\) be a partial linear space, i.e., two distinct lines from \(\mathcal L\) have at most one point of intersection. A subset \(X\subseteq S\) of points is called a subspace if every line that joins two points is contained in \(X\). It is called strong if any two points are indeed joined by a line. \(X\) is called a plane if it is a strong subspace, which together with the appropriate lines is actually a projective plane. The authors prove the following theorem: Let \((S,\mathcal L)\) is a projective or polar Grassmann space, such that maximal strong subspaces are at least planes and neither all of them are planes nor all of them are projective \(3\)-spaces. Then, \((S,\mathcal L)\) and \(\mathcal L\) with the coplanarity relation are definitionally equivalent. This means that every bijection on \(\mathcal L\), which preserves coplanarity is induced by a collineation of \((S,\mathcal L)\).