A note on isoparametric generalized quadrangles
An arbitrary isoparametric hypersurface with four distinct principal curvatures in a sphere defines a generalized quadrangle, the point and lines sets of which are the respective two associated focal manifolds \(\mathcal{P}\) and \(\mathcal{L}\). Incidence is defined by the point-line pairs that are projected from the same point of the compact connected hypersurface. In the paper under review, the author proves that two points \(p,q\) are collinear if and only if the difference \(p-q\) divided by the norm \(\| p-q\| \) belongs to \(\mathcal{L}\), and similarly for lines. Since collinearity completely defines the geometric structure of the generalized quadrangle (in particular, the incidence relation), this gives a simple algebraic description of the generalized quadrangle. For the Clifford type isoparametric generalized quadrangles, this result was previously proved by \textit{L. Kramer} [Geom. Dedicata 79, 321--339 (2000; Zbl 0958.51006)].
scientific article; zbMATH DE number 5077610