Veldkamp quadrangles and polar spaces

TL;DR

This paper explores the structure of Veldkamp quadrangles, establishing their connection to polar spaces and classifying flat Veldkamp quadrangles with specific relation properties.

Contribution

It provides a complete classification of flat Veldkamp quadrangles where some relations are non-trivial, linking them to polar spaces.

Findings

Connected Veldkamp quadrangles and polar spaces.

Classification of flat Veldkamp quadrangles with partial trivial relations.

Insight into the structure of Veldkamp polygons.

Abstract

Veldkamp polygons are certain graphs $\Gamma=(V,E)$ such that for each $v\in V$, $\Gamma_v$ is endowed with a symmetric anti-reflexive relation $\equiv_v$. These relations are all trivial if and only if $\Gamma$ is a thick generalized polygon. A Veldkamp polygon is called flat if no two vertices have the same set of vertices that are opposite in a natural sense. We explore the connection between Veldkamp quadrangles and polar spaces. Using this connection, we give the complete classification of flat Veldkamp quadrangles in which some but not all of the relations $\equiv_v$ are trivial.