Isometric embeddings of polar Grassmannians and metric characterizations of their apartments

TL;DR

This paper characterizes isometric embeddings of polar Grassmann graphs, showing they are mostly induced by collinearity-preserving injections, and provides a metric-based description of apartments in polar Grassmannians.

Contribution

It offers a new understanding of isometric embeddings in polar Grassmann graphs and introduces a metric characterization of apartments in these spaces.

Findings

Isometric embeddings are induced by collinearity-preserving injections.

Provides a metric characterization of apartments in polar Grassmannians.

Most embeddings of non-maximal singular subspaces follow this pattern.

Abstract

We describe isometric embeddings of polar Grassmann graphs formed by non-maximal singular subspaces. In almost all cases, they are induced by collinearity preserving injections of polar spaces. As a simple consequence of this result, we get a metric characterization of apartments in polar Grassmannians.