Geometric structures on finite- and infinite-dimensional Grassmannians

TL;DR

This paper explores the geometric structures of finite- and infinite-dimensional Grassmannians, analyzing relations like adjacency, distance, and lines, and their interdependencies.

Contribution

It introduces and studies the interrelations of various geometric structures on Grassmannians, including adjacency, distance, pencils, and Z-reguli.

Findings

Identified relationships among different structures on Grassmannians

Analyzed the properties of adjacency and distant relations

Explored the role of Z-reguli in Grassmannian geometry

Abstract

In this paper, we study the Grassmannian of n-dimensional subspaces of a 2n-dimensional vector space and its infinite-dimensional analogues. Such a Grassmannian can be endowed with two binary relations (adjacent and distant), with pencils (lines of the Grassmann space) and with so-called Z-reguli. We analyse the interdependencies among these different structures.