Interactions between para-quaternionic and Grassmannian geometry

TL;DR

This paper explores the relationship between para-quaternionic and Grassmannian geometries, highlighting their equivalence and interconnections in structures, normalization, curves, and twistor theory.

Contribution

It clarifies the equivalence between para-quaternionic and Grassmannian structures and links their geometric concepts and constructions.

Findings

Establishes the equivalence between almost para-quaternionic and Grassmannian structures.

Relates normalization conditions and distinguished curves in both geometries.

Connects twistor constructions across the two geometric frameworks.

Abstract

Almost para-quaternionic structures on smooth manifolds of dimension $2n$ are equivalent to almost Grassmannian structures of type $(2,n)$. We remind the equivalence and exhibit some interrelations between subjects that were previously studied independently from the para-quaternionic and the Grassmannian point of view. In particular, we relate the respective normalization conditions, distinguished curves and twistor constructions.