Projective structures and $\rho$-connections

TL;DR

This paper extends the theory of projective structures using $ ho$-connections in the complex analytic setting, leading to new insights into flatness conditions and applications to quaternionic manifolds and twistor spaces.

Contribution

It introduces a novel approach to projective structures via $ ho$-connections, enhancing control over flatness and enabling new geometric identifications.

Findings

Improved understanding of projective flatness through $ ho$-connections.

Application to quaternionic manifolds and twistor space structures.

Characterization of quaternionic projective space via complex projective structures.

Abstract

We extend T. Y. Thomas's approach to the projective structures, over the complex analytic category, by involving the $\rho$-connections. This way, a better control of the projective flatness is obtained and, consequently, we have, for example, the following application: if the twistor space of a quaternionic manifold $P$ is endowed with a complex projective structure then $P$ can be locally identified, through quaternionic diffeomorphisms, with the quaternionic projective space.