On orthogonal polar spaces

TL;DR

This paper demonstrates that for orthogonal polar spaces, certain geometric invariants called elliptic and parabolic gaps can be characterized intrinsically, without relying on their universal embeddings.

Contribution

It establishes that the elliptic and parabolic gaps are intrinsic invariants of orthogonal polar spaces, independent of their embeddings.

Findings

Elliptic and parabolic gaps are intrinsic invariants.

These gaps can be described without using the universal embedding.

The results apply specifically to orthogonal polar spaces.

Abstract

Let $\cal P$ be a non-degenerate polar space. In [I. Cardinali, L. Giuzzi, A. Pasini, "The generating rank of a polar grassmannian", Adv. Geom. 21:4 (2021), 515-539 doi:10.1515/advgeom-2021-0022 (arXiv:1906.10560)] we introduced an intrinsic parameter of $\cal P$, called the anisotropic gap, defined as the least upper bound of the lengths of the well-ordered chains of subspaces of $\cal P$ containing a frame; when $\cal P$ is orthogonal, we also defined two other parameters of $\cal P$, called the elliptic and parabolic gap, related to the universal embedding of $\cal P$. In this paper, assuming $\cal P$ is an orthogonal polar space, we prove that the elliptic and parabolic gaps can be described as intrinsic invariants of $\cal P$ without making recourse to the embedding.