Affine polar spaces derived from polar spaces and Grassmann structures defined on them

TL;DR

This paper demonstrates that affine polar spaces, as defined by Cohen and Shult, can be reconstructed from specific adjacency relations on Grassmann structures, generalizing previous results involving affine spaces with reflexive forms.

Contribution

It introduces a method to recover affine polar spaces from Grassmann structures, extending prior work on affine spaces with reflexive forms.

Findings

Affine polar spaces can be derived from Grassmann adjacency relations.

The approach generalizes earlier results involving affine spaces with reflexive forms.

Provides a new perspective on the structure of affine polar spaces.

Abstract

We prove that an affine polar space in the meaning of Cohen and Shult can be recovered from one of the three adjacency relations on a Grassmann structure over it. The result directly generalizes the results of our previous work where we use an affine space over a vector space equipped with a nondegenerate reflexive form as a starting point to the Cohen-Shult affine polar spaces.