Projective symplectic geometry on regular subspaces; Grassmann spaces over symplectic copolar spaces

TL;DR

This paper develops a framework for Grassmann spaces within symplectic copolar spaces, demonstrating that adjacency-preserving bijections correspond to automorphisms of the underlying symplectic space.

Contribution

It introduces a construction of Grassmann spaces for regular and tangential subspaces in symplectic copolar spaces and characterizes automorphisms via adjacency-preserving bijections.

Findings

Reconstruction of the underlying metric projective space from adjacency relations.

Bijections preserving adjacency are induced by automorphisms of the symplectic space.

Extension of classical projective geometry results to symplectic copolar spaces.

Abstract

We construct Grassmann spaces associated with the incidence geometry of regular and tangential subspaces of a symplectic copolar space, show that the underlying metric projective space can be recovered in terms of the corresponding adjacencies on so distinguished family of k-subspaces (geometrical dimension of the space being not 2k+1), and thus we prove that bijections which preserve the adjacency are determined by automorphisms of the underlying space.