Twistorial maps between quaternionic manifolds

TL;DR

This paper defines quaternionic maps between almost quaternionic manifolds and characterizes when such maps are twistorial, revealing conditions related to integrability and geodesic properties, with applications to quaternionic projective spaces.

Contribution

It introduces a natural notion of quaternionic maps and characterizes twistorial maps in terms of quaternionic and geodesic conditions, extending the understanding of quaternionic geometry.

Findings

Twistorial maps coincide with quaternionic maps for integrable structures.

Nonintegrable structures require maps to be quaternionic and totally-geodesic.

Application to quaternionic projective spaces elucidates map structures.

Abstract

We introduce a natural notion of quaternionic map between almost quaternionic manifolds and we prove the following, for maps of rank at least one: 1) A map between quaternionic manifolds endowed with the integrable almost twistorial structures is twistorial if and only if it is quaternionic. 2) A map between quaternionic manifolds endowed with the nonintegrable almost twistorial structures is twistorial if and only if it is quaternionic and totally-geodesic. As an application, we describe the quaternionic maps between open sets of quaternionic projective spaces.