Harmonic morphisms between Weyl spaces and twistorial maps II

TL;DR

This paper develops a unified geometric framework for twistorial maps and harmonic morphisms on Weyl spaces, characterizing these morphisms through twistorial structures and extending known results to specific dimensions.

Contribution

It introduces the notions of almost twistorial structures and twistorial maps on manifolds, unifying various examples of twistor spaces and characterizing harmonic morphisms in low-dimensional Weyl spaces.

Findings

Twistorial characterisation of harmonic morphisms between 4- and 3-dimensional Weyl spaces.

Description of twistorial maps with 1-dimensional fibres from 4-dimensional Weyl spaces.

Unified framework for known twistor space examples.

Abstract

We define, on smooth manifolds, the notions of almost twistorial structure and twistorial map, thus providing a unified framework for all known examples of twistor spaces. The condition of being harmonic morphisms naturally appears among the geometric properties of submersive twistorial maps between low-dimensional Weyl spaces endowed with a nonintegrable almost twistorial structure due to Eells and Salamon. This leads to the twistorial characterisation of harmonic morphisms between Weyl spaces of dimensions four and three. Also, we give a thorough description of the twistorial maps with one-dimensional fibres from four-dimensional Weyl spaces endowed with the almost twistorial structure of Eells and Salamon.