Twistorial structures revisited

TL;DR

This paper revisits twistorial structures, providing a framework involving $ ho$-connections to describe their differential geometry, offering new proofs and insights into projective and quaternionic geometries.

Contribution

It introduces a setting with $ ho$-connections that simplifies the understanding of twistorial structures and their associated geometries, including new proofs of existing results.

Findings

Established a unified framework for twistorial structures using $ ho$-connections

Provided new proofs for the existence of relevant connections in projective and quaternionic geometries

Demonstrated that the Ward transformation follows from $ ho$-connection properties

Abstract

We review the twistorial structures by providing a setting under which the corresponding (differential) geometry can be described, by involving the $\rho$-connections. This applies, for example, to give new proofs of the existence of the relevant connections for the projective and the quaternionic geometries. Along the way, we show that, in this setting, the Ward transformation is a consequence of the good behaviour of the $\rho$-connections, under pull back.