Projective structures and $ρ$-connections

Abstract

We extend T. Y. Thomas's approach to the projective structures, over the complex analytic category, by involving the ρ-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.